Installing trap with new deeper sink
The normal solution to this is to trim the lower end of the black part.
Tailpieces are supplied long, and expected to be trimmed to fit. Of course, there are limits to that.
In this case, it looks like the elbow in the black part is going to cause a problem (or be a limit) with that approach. Such is the danger of a deeper sink. You may indeed need to lower the drain entrance into the pipe in the wall, unless you can find a disposer that exits a little bit higher (which would be easier, and possibly cost less.)