If you have leftover liner, you can use that as your patch. Then it's a matter of finding what works for waterproofing the seam. When I sealed my hole (I think it was from a turtle digging in my bog, 3 slices in a row), and since it was on the bottom, I had gravity working with me. I had to pull all the bog pea gravel out, so pulled the liner back, dried and cleaned it with acetone, then roughed it up with sand paper. I had bought a patch kit for a swimming pool, got the biggest kit for tear I could find, so would have more glue. Then I also used the PL Roofing stuff that comes in a tube like caulk. It looks more like roof tar when it comes out.
I put a piece of pond liner patch down, then the glue and PL stuff, then the old liner with hole. On the inside of the bog bottom, I also added another patch, doing the same thing. I didn't mix the two glues, but rather used one on the inside of the circle, and the other on the outside. My thought was if one failed I had the other as a backup. Then on the inside, I added a second much larger patch of liner material with just the PL Roofing stuff. No more leaks that I know of. Oh, and since I was in there, and since I had extra liner, I put a strip of extra liner down on the bottom of the bog floor as extra protection as well.
In your case, make sure the patch is on the inside wall, I would think anyhow, using the water gravity to hold it tight. I don't know how that will work, though, if your new hole is right above the old hole, maybe part of the old hole. I'm giving suggestions, but hoping someone that has done more with patching or knows more of the elements that need to be used chimes in. I just found that the PL Roofing stuff worked the best to seal anything and everything around my pond, from bottom drain on Skippy filter, to this pond liner leak.
Wondering ... since you said that the water level goes down just to the level of I assume the bottom edge of the skimmer opening, is there any way for you to remove the rocks on top of the skimmer, and look down into the hole where the skimmer is placed, and see if that is in fact the source of your leak? I would think you would first want to find out for sure where the leak is before you do the patch and skimmer moving, although I think you want to raise the skimmer either way ...
Good luck with your issue.