The trick is to use your palms on that last part that needs to go in. Press your palms against the tire, loop your fingers around the top of the wheel, and apply pressure so you are rolling your palms up and down/back and forth. I'm not a whiz when it comes to other parts of the process, but I have helped more than a few experienced riders who were having trouble with this.
The Pedro levers are the best, but I've never used them for putting the tire back on. And some tires are just easier than others to work with. A friend just asked me why I didn't have a certain brand of "flat proof" tires and I responded that if I did get a flat, I'd never be able to get those tires on or off. I've found Contis or Shwalbes are much easier than others.
2015 Trek Silque SSL
Specialized Oura
2011 Guru Praemio
Specialized Oura
2017 Specialized Ariel Sport