If you put an opposite word of an adverb , it does not convert the word into the opposite meaning.
slow down cannot be turned into slow up to mean faster!
The word you may use is flared which means widening up.
Though it has limited usage and may not suit to your concern.
ell.stackexchange.com/users/3187/Maulik V
You wouldn't get the meaning you're looking for if you said 'taper up'.
The immediate thing that comes to mind when I hear 'taper up' is a tall building 'tapering up' toward the sky.
In this case the adverb is referring to the direction that the thinner end would point if it tapered to a point.
You could also say 'tapered to the right' for instance.
That doesn't mean that 'tapered down' always means that the tapered end points down, though. '
to taper down' is a verb phrase that means the same thing as 'to taper'.
The word 'down' in this case is not referring to the direction, but to the width getting smaller or lower.
It would be perfectly acceptable to simply say "became gradually thicker toward one end"
ell.stackexchange.com/users/74475/dwilli
A baseball bat broadens out above the handle.
ell.stackexchange.com/users/10264/Tᴚoɯɐuo
To taper just means to become thinner as the length increases, or as you move along the length.
down
is only used as a direction, it's not a phrasal verb .
There is a phrasal verb to taper off , but it has a different meaning: to diminish over time, not over space.
Both wider and thicker (in your question) have related verbs: to widen to thicken However, to thicken has connotations of food: to thicken the sauce .
To widen looks like the word you want.
The beam widens the further from the source it gets.
ell.stackexchange.com/users/9936/CJ Dennis
First, it's most usually used with "off", as in "taper off".
"Down" would be valid but a distant 2nd.
The nearest and most popular antonym to "taper down" would be "spread out", followed by "fan out", "enlarge", or "thicken"
ell.stackexchange.com/users/74981/Jason
In what context?
Gradually reducing something (like doses of medication) over time, is often called tapering down, (sometimes, tapering off), so logically we might expect gradually increasing something to be tapering up.
Oddly, I've never heard that.
However, "ramping up" is widely used in the context of gradually increasing something, for example, Tesla is ramping up production of its new car.
For something physical you can simply turn round and look at from the other end, I think "taper" is OK
but I agree "taper up" is a little strange, perhaps "widen" or "thicken" at one end is better.
ell.stackexchange.com/users/12606/Brian Drummond
AS your question already hit on, you're looking for the word "widen."
ell.stackexchange.com/users/48180/fectin
The verb "taper" is defined in dictionaries as:   To become gradually  narrower or thinner  toward one end.   So, this word is often used with the adverb "down".  I am searching for a word that means To become gradually  wider or thicker  toward one end. Is it possible to use the expression "taper up" for the meaning?
Is the expression "taper up" incorrect?
ell.stackexchange.com
<verbs>
ell.stackexchange.com/questions/165737
ell.stackexchange.com/users/46924/rama9
If you put an opposite word of an  adverb , it does not convert the word into the opposite meaning.   The meaning would change if you said 'taper up'. There is a phrasal verb  to taper off , but it means to diminish over time, not over space.  "Flared", "spread out" or "fan out" may be used for "widening or tapering up". While "taper down" or "taper off" are used, these are phrasal verbs and cannot be reversed in this way. Acceptable antonyms could include "flare", "spread out", "fan out" or "thicken".
While "taper down" or "taper off" are used, these are phrasal verbs and cannot be reversed in this way. Acceptable antonyms could include "flare", "spread out", "fan out" or "thicken".