If I use a garlic press, there is a remnant of crushed garlic which needs to be disposed of. However, I usually find that this then makes the whole bin (trash can) smell of garlic until I empty it, which can be several days to a week later. (This isn't an issue with the smell lingering on other items; once the bin bag is gone the smell is gone too.) Is there anything I can do with the garlic remnant to stop this from happening? 
One option is to stop using the garlic press.  This is a tool that results in the waste that you describe, plus it takes time to clean after use.  Instead, just use a knife.  With a chef's knife, for example, you can slice, chop, mince, or crush (more options than a garlic press!).  The advantages are many.  First, you can prep your garlic in a variety of ways.  The flavor and aroma garlic adds to a dish can be greatly controlled by how it is prepared.  Then, you only have the peel to dispose of, and the knife can simply be washed off. 