The noun: a first approximation or the first-approximation solution
This means a rough initial solution that doesn't solve the problem precisely.
The verb: make a first approximation
This term comes from math, more specifically numerical methods, but people who understand math use this all the time for non-math problems and partial solutions. I hope you find it useful too.
I'll explain the concept in math. You have a problem for which an exact (pencil and paper) solution does not exist or is too tedious to search for. So you write a computer program to find an approximate solution. Your algorithm is iterative, meaning you start with a rough approximation and then apply the algorithm repeatedly, to bring your estimated solution closer and closer to the true solution (which you can probably never know with infinite precision). After going through the algorithm (procedure) once, you arrive at the first approximation [solution]. After a second pass, you arrive at the second approximation. Etc.
Before computers, numerical methods could be used but it was tedious. So people had good reason to keep track of exactly how many times they went through the procedure.