There are tons more techniques for escaping mount, but the ones you have been taught are the ones you should focus on.
There are several reasons for this.
Focus on fundamentals
The two escapes you know are arguably the most straightforward and efficient methods of escaping mount.
In addition, they both develop absolutely critical grappling movement skills : the bridge and the shrimp.
These two movements will be used in all sorts of situations.
They are also complementary: if the person on top stops one, the other is often easier to apply.
Instead of looking for other techniques, you'd do well to put your effort into applying these two techniques against a variety of opponents.
The other major reason to stick with these is that they are high percentage techniques.
Even the best jiujitsieros have trouble with these two escapes.
They are used in elite BJJ competition.
Don't discount them.
Don't be a YouTube grappler Aesopian, a BJJ black belt who has been prolific on the web since he was a blue belt, used to be a voracious reader/watcher of instructional tapes and books.
He has this to say about learning in this manner :
You find some interesting techniques in a book or online.
You glance them over and make a note to try later.
At class, you’ve got your attention split between what is being taught in front of you and the half-forgotten tutorials floating around in your head.
When sparring comes, you drop whatever the day’s lesson was and fumble to piece together something else entirely.
He goes on at length.
This is the fate of the grappler who attempts to learn technique from the internet or instructionals, instead of in class.
Don't learn your technique from YouTube.
Learn and drill in class.
(That being said, learning techniques from video can be marginally useful in some limited sense .)
Improving your upa
The bridge is a fundamental movement not just to escaping mount, and not just to Brazilian jiu-jitsu, but to all of BJJ, all grappling, and all fighting.
Getting your bridge to be technically efficient as well as powerful is a great goal no matter what level of jiu-jitsu you have.
When I'm working on my upa (bridge), I work on strength, power, and technique in the following ways:
(Technique) Movement drilling - I practice the movement on my own.
I make sure I get my feet close to my butt
, I make sure I'm not exposing my arm
, I make sure I'm extending my hips fully
, I make sure I'm pivoting on my shoulder.
There are a variety of movements (Strength) Barbell hip thrusts -
I hope the implications of being ferociously strong in this movement do not need to be explained.
Other general strength exercises that assist this movement are the barbell squat and deadlift.
(Power) Olympic lifts
- The barbell power clean is a tremendously successful method of translating strength gained from other strength work into speed-strength (power), which is the most relevant attribute to athletics.
(Technique) Positional drilling - The simplest way to get gobs of training time in mount escapes is to drill it repeatedly.
Give your partner mount in free rolling, or outside of class get a skilled partner on top of you.
Set the timer for ten minutes and switch every time you escape.
Have them resist, and try to submit you, to the best of their ability.
Do this with as many partners as possible, and you will get very familiar with the minutiae of what works for escaping.
martialarts.stackexchange.com/users/347/Dave Liepmann
There are other options, but bump and roll and elbow escaping are the best to spend your time developing.
One tip that helped me a lot early on was learning to combine those two techniques effectively.
Use a failed upa to set up an elbow escape.
There have been some good tips foraging a more powerful bridge already listed, but I think most people have more of a problem making sure the leg and arm are properly trapped than they do actually bridging.
martialarts.stackexchange.com/users/666/Josh
You can escape out the backdoor - underhook their legs with at least one arm and rotate to your knees (quickly, or you'll get triangled).
You can press them away and tuck both knees in to get butterfly guard.
If they have high mount you can reverse figure four them.
You can overhook an arm and bridge over the other shoulder.
If you have good choke defense you can give up your back, as long as it's going to your knees and not your stomach.
You can bait the armbar and do a hitch-hiker escape.
You can bait the americana to facilitate an easier bridge and roll.
To get a more powerful buck, get your heels to your butt first.
The further your heels are away, the weaker it is.
martialarts.stackexchange.com/users/565/Robin Ashe
Here's something I liked to do when doing bjj, being a flexible guy:
If the mounting guy sits up, I would put my leg between his torso and mine, kind of like when rubber guard, but in front of him, then it's really simple to push him away since your legs are so strong.
If he's isn't sitting up, i.e., making an s-grip with your hands between his ass, bridging and turning him over worked well, but requires a lot of explosiveness.
martialarts.stackexchange.com/users/1265/Mårten
I use bridge and roll if they are over my hips but its not all that effective in my opinion.
If they are very high up
I elbow escape\shrimp to get my hips lower and then use either of these 2 escapes: 1) Turn on my side, frame the hip with both hands, push their knee wide with my bottom knee exposing their shin, hook the shin\ankle with my top leg, using elbow and the hook lift their leg and get half guard.
I have been using this since white belt, its effective for getting back to half guard and I reckon your instructor should have shown this.
2)
Shrimp hip escape and get full guard
Those are the only 2 escapes I personally use, combined together to get half or full guard.
I sometimes use the bridge and roll to create some space and get onto my side.
martialarts.stackexchange.com/users/7823/berimbolo
Elbow Knee escapeRoy dean has some great techniques for escaping the mount!
check out his blue belt reqs dvd
martialarts.stackexchange.com/users/1475/Funky
I know the two most fundamental escapes: the "buck and roll" ( upa ) to and the shrimping knee-push guard recovery (in which you buck onto your side, push down on knee to get half guard, and work from there for full guard).  Are there any other options within context of BJJ?  Are there any excercises for producing a more powerful buck?
What are some options for escaping full mount in BJJ?
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<brazilian-jiu-jitsu><escapes>
martialarts.stackexchange.com/questions/1322
martialarts.stackexchange.com/users/572/daniellopez46
There are many techniques for escaping mount, but the ones you have been taught are the ones you should focus on. Learn and drill in class. Improve your  up,  Olympic lifts, and positional drilling. The simplest way to get lots of training time in mount escapes is to perform drills . Bump and roll and elbow escaping are worth developing. Most people have a problem making sure the leg and arm are properly trapped than they do actually bridging.  You can escape out the backdoor - underhook their legs with at least one arm and rotate to your knees. You can press them away and tuck both knees in to get butterfly guard.  If they have high mount you can reverse figure four them. If you have good choke defense you can give up your back, as long as it's going to your knees and not your stomach.  You can bait the armbar and do a hitch-hiker escape. You can bait the americana to facilitate an easier bridge and roll. There are many techniques so  focus on the ones you have been taught in class or you can use  bump and roll, position drilling, creating a more powerful bridge, olympic lifts, pressing them away to move into butterfly guard, and baiting the armbar and americana.
There are many techniques so focus on the ones you have been taught in class. Examples of techniques you can use are bump and roll, position drilling, creating a more powerful bridge, olympic lifts, pressing them away to move into butterfly guard, and baiting the armbar and americana.