It is definitely a tough challenge to have. My advice would be to start very early and be strategic on who you chose. I was applying to roughly 10 schools and knew that I was going all-in this year and did not want to re-apply. I had a rough pool of about 10 recommenders, and about 3 to 4 who knew me very well and who I knew would submit the letters very timely. I basically divided up the recommenders to different schools so they would only have to do about 3-4 schools each.
Since July/Aug is usually when most apps come open, the recommendation letters are there for them to submit. Since 1st round due dates are not usually until Sep/Oct/Nov, I would start right now if not much earlier.
I basically started to reach out during May/June timeframe and gave my recommenders until end of August to submit. I finished letter submissions for round 1 by Sep for all recommenders and my first round 1 app was due in Nov, about a two month window. I document my application process here if you would like more detail:
https://3hrmbaapp.weebly.com.