Why would each patch produce a new leak before resintalling tube? I got a puncture in my tube (size 700c x 23c, Kendra, came with bike) and applied a vulcanizing glue and patch, then checked it. Before reinstalling the tube, there was now a new leak right next to the patch, but the air is definitely not escaping from the lip of the patch, just very close to it.

So, I added another patch. Now the same thing happened again: a new small leak right near the patch but not from the lip of the patch.

I noticed these new leaks before reinstalling the tube, so the problem is not anything on the rim or in the tire.

I am relatively new at this so wanted to see if this has happened to others and what might be causing it.

Following the patch kit instructions, I applied pressure using the kit box from the center of the patch out. Would this potentially rip a new hole? Do I need to be very gentle with this step?
You may have a sort of systematic effect, such as debris (puncture causing material) near the patch site, or missing rim tape or something depending on where the patch is. It may not be easy to find the causing material unless the tire is flexed under load though, which is why you might just not see it.

Either that, or you're consistently not applying the patch right (which you just need to follow the directions in the patch kit).