Can wearing a hat everyday 1 month after a hair transplant affect future growth?

Wearing a hat will not cause hair loss unless you wear your hat is so tight that it is causing traction alopecia.

You are perfectly safe wearing a hat after transplant, particularly one month out.  You should not have any problems with future growth due to wearing a hat. 

In my practice after hair transplant surgery I give my patients a special hat to cover their head when they are outdoors. It is a light and comfortable head without making pressure on the recipient area.I also advice them that they can use any hat two weeks after their hair transplant surgery.On the other hand I mention not having helmet or similar heavy hats for one month time post operative.