Problem is, by going with one guy, he has to be at the right place, and able to head in the right direction, at the right time... Very small chance of that.
If you call the big places, you just have to be a little pushy. Basically call a few, and tell them you will only book with them based on the stipulation someone picks up the car within 24 hours (or 48 if you aren't in that big of a hurry). It's amazing the things they can make happen when they will lose the business otherwise.