I kept a good sized cup in my cup holder, which lasted most of the drive. Remember, the volume you wind up spitting out in the cup will be more than you first put in your mouth. I also kept a bottle of water in the second cup holder.
There is no need to rinse between pulls if you're doing back-to-back pulls. But definitely rinse if you are going to stop for a while (like to eat, make a "pit stop", or just need a drink of water). That's what the bottle of water is there for. Use it to rinse when you need to, and to get a drink if you're thirsty.
I noticed on my drive that after about pull 4 or 5 the pulls got thicker quicker for me and lasted longer until thinning, so I could tell I needed the earlier pulls to just jumpstart things. I could time it precisely by pull 4. The first few didn't seem as effective (I've been pulling for a long time, so I get results much slower anymore), and took a longer time to get results. I used Walnut Oil for that trip. It was convenient because it had a flip top with a small hole for squirting the oil, so I didn't have to worry about accidentally spilling down my shirt when dosing for the next pull if I hit a road bump (a good squirt was about equivalent to a tablespoon, so I didn't have to do any real measuring ... squirt in a comfortable mouthful, then start swishing).