I can tell you who not to use...."Anything Around The House" in So Austin. Wife had a recommendation for them. I had a bad feeling, but she was going to be the primary person dealing with them. Half way through the job, I had to get in the middle of it. Out of the 10 or so things we had them do as part of a small / medium remodel, 9 had issues.
The challenge is that there is so much building and large remodels going on that the good talent is focused on that work. Most of the remaining handymen are the scrubs, but I'm sure there are a few good ones out there.
If you have a good relationship with a realtor you may ask them for a recommendation. The handymen want to keep those relationships strong because the get a lot of work from realtors after inspections.