Leave in April and working clockwise so south in early May and north in June. Be out of Yellowstone by end of June.
You don't say how big your rig is. The bigger the more reservations will help. The smaller, the more you can get by without. However, I almost always travel without reservations. Trick is to be ahead of the crowds which generally means before school lets out. Then be at the busier places during the weekend, especially avoiding them on Saturday night which is usually the hardest to get. Be at the smaller campgrounds non-NP campgrounds for the weekend when necessary. My backup plan is I can boondock for a couple of nights if necessary.
Another trick I use so my schedule isn't locked up is I only make my reservations out for the next spot. Say I'm in location A with a plan for 4 days. On day 2, I'll have a better idea how long I want to be at location A. So then I can call location B to try and make reservations. May have to either come up with spot C for a couple days or stay at spot A a couple days longer. Of course, the first question I ask before making the reservation is how many sites are open. If the campground is wide open, no need to make reservations (unless you have a very large rig).
Granted it hasn't been in the last few years because I'm still working, but I've been lucky to travel the country twice with very limited reservations. The last trip was 8 weeks long with a 35' FW and I had reservations at Grand Canyon Trailer Village for Memorial Day weekend and Yosemite a couple weeks later. We still managed to hit Bryce (Ruby's Inn just outside), Zion, Redwood, 6 state parks along Oregon Coast, Glacier, Yellowstone (made reservation 2 days out), Badlands, and bunch of other little spots. Turns out we were 1 week ahead of school in Oregon and Glacier was still mainly snowed in but Going to the Sun road was open and campground was empty.
I retire in a couple more years and when we start traveling again, we won't have any reservations as no more schedule for us. Of course, we now only have a small truck camper so only need a somewhat level parking space making spots to come by easy and backup boondock options plentiful.