It looks like a jack was put in the wrong place and bent the cover lip up...alternatively, the car was driven over something high enough to bend the cover back, but it really looks like it was "lifted" from below. In any event, you might first try tightening the cover bolts a bit---don't overtorque them---to see if it stops the leak (if it does, top your diff oil off). If not, covers can be had fairly cheap from salvage, or from aftermarket suppliers. With that in mind, if tightening it doesn't stop the leak, there is no reason not to work it a little with a hammer to try and get a straight edge for a seal(use a straight edge to judge your work...ALSo, check to see if the cover is cracked. The hardest part of this job for me was figuring out a good way to get the oil back in it...a lot of obstructions on the front side, especially when trying to get the diff level...I finally fell back to a piece of hose jammed on the gear oil bottle nozzle. Note that OEM was put on with silicone gasket goo (no gasket), so when you put a gasket in there you get some extra crush that will help seal your diff if the cover is a tiny bit off. ((And diff oil stinks...try not to get it on you)). Have fun!
Edit...reread your post...get the car safely on jack stands...basically put a large catch pan under the diff cover area and loosen the bolts enough to allow the cover to separate from the diff to let the old oil out. There is a 1/2" square head plug on the driver's side front---the oil goes back in there once the cover and new gasket are installed. Try to get the car/diff level and fill the diff until the oil is even with the bottom of the fill hole (it will seep out when you are there).