The previous owners had the floors screened and re-coated with one coat of oil based poly. They look great except for the living room. The previous owners said that the flooring guys had a hard time with the area and that their screens were getting clogged so they went to lower grits but I guess they rushed the job since they didn't use higher grits to get all the swirls out.

We bought the house and I found some used 80 grit screens. Anyway, the floors are in excellent condition, we love the color but want to remove the swirls which can be seen when the room is well lit or you look for them. I was thinking of having a flooring guy come in and re-screen the floors starting with 100, then 150 and 220. Then re-coat. Does this sound like a good idea. We were quoted a price to sand them completely down and it was way out of our budget. Thanks in advance

The directions for a re-coat on the cans of poly-urethane that I use suggest 100 grit screens. Sharp 100 grit screens will create tooth for the finish adhesion. A problem I have seen is that one coat of finish over this preparation will not always cover the screen marks since it takes a lot of practice to apply an even coat following the suggested application rate without leaving "rivers". Of course applying too thick in one area with the correct spread rate will result in thin spots elsewhere which may show up as screen marks in reflections.
I find that 2 coats applied a day apart from each other results in the best results.80 grit screen marks will be more difficult to cover, but applying another coat as soon as possible will help guarantee adhesion as the freshly screened coat will help the final coat bite in without a lot of abrasion from the screening of the relatively fresh first coat.
If you coat again before the last coat has completely cured, you should be good screening with 120 grit.