I'm 3.5, and I've been working on my game. My results suggest I'll move to 4.0. So I'll weigh in on what has helped me improve.
First, I would suggest getting a devoted practice partner. I hit pretty regularly with another woman who is trying to improve. This gives me a chance to practice what I am learning in lesson. We practice between 2-5 hours at a time. Warm up. FH crosscourt. BH crosscourt. Volley-to-volley. Baseline to volley. Crosscourt rally moving toward the net. Always finishing with overheads.
Second, I think it isn't easy to improve without instruction. You mentioned taking "a lesson." Maybe a lesson will help, but honestly if you can only take one lesson it won't make a difference IME. Maybe find a friend and do semi-private lessons to keep the cost down?
Third, I got huge benefit when I started drop feeding instead of hitting against the wall. I find that hitting against the wall makes the ball come back before you are ready. So you take short cuts to hit the ball back -- no shoulder turn, no takeback, no follow through. With drop feeding, you can take your time and get some muscle memory going without the pressure of the ball machine or the wall.
Good luck. Moving up isn't easy.
-- Random Error Generator, Version 4.0
-- Master Moonballer