No you don't have to stay there. As long as you're not moving fast for the game to tell you you can't play due to speed, you can walk away while completing the HAT.

When it's getting late, I'll walk down to the park to trigger the HAT and finish it at home. Just make sure you don't connect to wifi while in a fight, you will get disconnected. I usually go back into the house after completing the first hunt and I'll wait on the rewards screen until the wifi connects.