Best way to avoid that is to make your transfer paper larger than your hat press, so that it doesn't have edges. Which means printing less up which is a bummer.
Or to put some larger paper down like Eric said, ideally that goes outside of the heat patten area.
Or you can cut your transfer close to the image which you did, which is pretty unrealistic for larger runs.
The pressure mark may be able to be steamed out, but they usually go away with just regular handling. We have some picky customers that think the little lines are deal breakers, so we print less up per transfer so they can be cut so the paper is larger than the platten, which then sort of tapers out the effect from the heat press on the hats since pressure isn't even across the presses.
Could also be from your paper, is your paper picking up any of the black from the hat after pressing? We use Arjo Wiggins T75 which we hot peel, and Burkhardt Freeman
http://burkhardtfreeman.com/polystrip_wb.html which is cold peel.
