Personally I'd rather get a few more bids then make a decision.
Then why are you rushing into this? I'm assuming that you work hard for your money - why spend it without knowing exactly what you'll be getting? Talk to multiple contractors and people that have had the same type of work done recently. Talk to the local building department. Is he licensed and bonded in your town? What's his track record with the town? Is he getting the right permits (if your town requires them)? That isn't something that you can usually walk in on tuesday, get the permit while you wait, and then do the work on friday.
A quality contractor might leave a flyer and business card on your door if he's in the neighborhood but he isn't going to start trolling for more work just because he's there.
A quality contractor also isn't going to give you a deadline to get the work done by. He might give you a dead line that pricing will be honored until (asphalt is very volatile with the price of crude oil). Doing the actual work would fit into his work schedule.
Personally, I'd rather take the time to research contractors, the product, and pricing before agreeing to anything with anyone. I'd also be willing to spend more money to get a quality product, installation, and warranty from a reputable company.
Like anything else in construction, the final product is only going to be as good as the prep work. If the contractor isn't doing any prep work, he isn't doing you any favors. In fact, he's hurting you because when the pavement fails in a couple of years, it will have to be torn out at your expense.
Coming out of winter, there are bound to be soft spots in the gravel that need some work. The whole driveway also needs to be graded to avoid puddling and to make sure that you have positive drainage away from the house and garage. Is he compacting the base before he starts paving? He should be. Small jumping jacks and plate compactors don't count except right alongside a structure that a vibratory roller won't physically fit.
This whole subject has so many red flags flying right now, you'd be crazy to let him set foot on your property let alone do actual work. No offense but if you make a lot of rash decisions like this, I can see how you arrived at your screen name.