Sump, French drains, or other?
I wouldn't think that even in bad conditions the water table rises all that much, I'd guess your infiltration is from the side walls as rain drains down into the earth. Re-grade next to the foundation. 

If that doesn't help, then yes, you need a real sump. I'd need a pic to say if I'd make a new hole or not. French drains can be added later, but you'll need a sump regardless. Or not, if re-grading takes care of it. Order of attack:


Re-grade next to the foundation  
Add sump  
Add French drains


If you're serious about keeping it dry so you can use it as a rec room or whatever, arbitrarily do all three steps.



Seal the foundation with some caulk and paint it. Or skim coat the whole thing and then paint it. At the very least, some paint will show you where the trouble spots are once they discolor. If there is space permitting, I'd leave that old pump alone as a backup, and dig a new pit if you're going to put a new sump in.

But if you're 5~6 feet above a river, a real sump pump might be a losing battle... Caulk the wagon and ford the river.