This sounds sort of familiar. Except for the "I am not talking about a flood" part. The basement here fills with about 1.5" of water when it rains a lot. It only reaches that depth because at that point it just starts flowing out the back door. Water comes in through:
2 cracks in the front of the house's foundation
1 crack in the back of the house
The sump
The corner clockwise from the pump
Cracks in the floor
The chimney
The damn place is like a piece of cheese cloth. The last time it rained hard here, at the end of June, we got 6.5" of rain in 48hrs. The sump pump was running continuously, but it just wasn't enough. It's rated 1/3 horsepower with a 1.5" diameter outlet. It still wasn't enough. And it wouldn't have mattered much, since water came in just about everywhere else. The ground slopes toward the house on 3 out of 4 sides too.
The water coming in the front of the house....well, if you're standing in the basement looking at the one crack in the wall, the water can come in at a level about 6' up from the floor = about ground level outside.
What we did:
A contractor came and gave an estimate, and we liked it, so we hired him. He dug a trench to about a foot or so below the bottom of the foundation (10' deep total), from the side of the house around to the front, put a 4" perforated pipe on the bottom, filled a few feet above that with grade 2b crushed stone, put some filter fabric over top, and then filled the rest with dirt.
In addition to that, he did a "core drilling" round the back of the house, in the corner where the water was coming in. As he explained it, the foundation walls go down a few extra feet as a frost barrier. However, this can trap water. Underneath the basement floor is a bed of crushed stone, which is where the water is trapped, so it becomes an underground lake, and its easiest way out is up through the floor. He drilled a hole in the frost barrier at the bottom of the crushed stone layer, and put a pipe into there and cemented it in place. That pipe joined with the one from the trench, and the whole system now drains into a drainage ditch between our lot and the neighbor's property. His work cost about $4,000, which was several hundred less than the estimate. In addition to that, I rented a backhoe for two days and did some other lighter digging at a crack in the foundation in the back of the house, as well as some land grading. Total cost of the project is close to $5,000
I also have no idea yet if it works. It's hardly rained at all since the work was done. Everything in the basement is staying up on 2x4 blocks until we're sure it won't flood again.
Excavating around the entire perimeter of the foundation for water sealing was not feasible, as it would have meant destroying/removing: part of the paved driveway, a flowerbed, a deck, and two porches. So the trench was only dug immediately next to the house on just one side. Around the front, it is several feet away from the house, but that should provide an adequate diversion for the water streaming down the nearby hill.
In any case, enjoy your comparatively dry basement.
It could be worse. In the same storm that flooded our basement with 1.5" of water, some people got more than 2 feet in their basements.
Your other option: rent a backhoe for a week and dig out the foundation yourself. This of course depends on how well you can get acclimated to hydraulic machinery. For me, it took about 2hrs to get used to the controls, and by the end of the first day I could reliably guide the bucket to dig to within an inch of the foundation. Just be sure to do something to shore up the walls, or else slope them, to avoid cave-ins. OSHA says that a trench more than 4' deep sound be reinforced or sloped. If that much rocky dirt caves in on you, you'll probably suffocate before anyone could dig you out.
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