As others have stated you can do some things to get a better finish while turning. I may add using a good cutting fluid with HSS can also help. But fluid is not recommended with TC tools.
I also make some things from mild steel. Cold rolled, hot rolled, 1018, etc, It is cheaper and easier to come by. I keep some rolls of emery cloth or wet/dry paper strips at the lathe. Tear off a 6-9 inch length and put a few drops of oil or cutting fluid on the work and apply the emory cloth like a shoe shine cloth. About a half wrap around the work and work it back and forth for several inches along it's length. Work it back and forth along a section of the workpiece. Start with about a 150 or 180 grit for a half minute or so. Then a finer grade like 320. Then 500 or 600 and finish off with a 1000 or 2000 grit. Be sure to clean the work off between grits. Use a speed that is one or two steps higher than the turning speed for the metal and diameter. You should be able to get close to a mirror finish on the part in a minute or so.
I have finished a lot of parts this way. It removes only about 0.001 to 0.0015 inches on the diameter for mild steel.
I do not use this technique on sliding or bearing surfaces as some grit may remain embedded in the finish and cause accelerated wear. For that kind of work, I just bite the bullet and purchase a better grade of steel.
Make it fit.
You can't win and there is a penalty for trying!