It seems like they built a bunch of M3Ps either right now or just recently. There are still some folks waiting, so we'll have to see whether they keep building until all the current orders are satisfied. It seems like the last batch of M3Ps built was in March. So on that basis, you'll want to get one of the ones in the current batch if possible, or else you might be waiting as much as 4-6 weeks for the next batch. Though, Elon did talk about "unrolling the wave" on the Q1 call, so maybe they will try to break it up a bit more? Hard to know.
I did learn that it's worth calling a service center regularly, and having them check their inventory, which shows more options than the public inventory site. Each day I called they turned up a car, though the mileage and colors varied. I ended up getting a delivery date on my original order before taking one of those, but it was great to have options (other than "wait in silence").