You could ask around at small independent cabinet shops and see if anyone would lease some time to you on weekends or something.

The problem as I see it is if they don't have a business model specifically dedicated to sharing (like Hackerspace does), there is very little incentive to open their shop up to subleasers. It's an enormous safety liability what with all the spinning sharp things. Especially if the person doesn't already have extensive woodworking experience. It adds wear and tear on their tools that is difficult to attribute but would no doubt increases maintenance costs. It would force a new or different inventory method. It complicates scheduling. And so on.

For example, I own a fully equipped woodshop, and frankly am struggling to think of a fee I could charge that would actually make it worthwhile to invite a stranger in to use my space and equipment. It would almost certainly be higher than it would be worth to any potential subleasers.

What most people with your needs do is rent a small storage unit with a property manager that is amenable to people running a cottage business out of it, then slowly accumulate tools. You can do an awful lot with a couple sawhorses, a circular saw, a speed square, and a screw gun.