Is it safe to wear a hat while working out, or will it damage my results?

You can wear a hat after surgery starting from day 1.  So after one year, you should be able to wear a hat.  

Wearing a hat after the 3rd week post transplant is safe. During the first 3 weeks one can wear a hat, provided it is gently put on and worn properlyAt the time you have stated (one year post op) there are no problems wearing a hat - whatsoverIf growth is not as expected, speak to your surgeon

Hats do not affect it this far out but like all medical procedures no results are 100% in all patients.  First, wait the full 18 months, then talk to the clinic about a repeat procedure.  Sometimes the second round works and often it is much cheaper.  Not all scalps work well all the time the first time around.  No legitemant doctor can claim that. 

At one year, you should see most of all of the results of a hair transplant. You can do whatever you want with regard to working out. If you have a problem with growth, go back to speak with your doctor and ask questions about his experience with failures to grow and if you have that. Were your expectations set correctly?  See what he will do about any failure. 

It is safe to wear a hat. You can return to your normal routine after three weeks and that includes wearing a hat unless it is too tight and pulls your hair.