At a rental we had, i tore a 2" hole when the motorcycle I was parking (newbie) fell over and the handle caught it.
It was about 1/2 inch from the edge... but -- i just took a needle and black thread and darned it like a sock. Was not overly noticeable and the landlord did not deduct for it even through they were especially picky about the place on our departure.
If it was my own screen door and I was especially cheap -- i think I would try the same, but wrap the thread AROUND the frame, maybe, to anchor it where it is too close?
Of course, I would try that because it is likely I will rip 3 new holes before the end of summer, and I want to wait until it is ruined before spending for the replacement screen (of which I, too, seem to have an extra 50 feet of, why do they only sell in such large rolls?)