The K-Mets trail Greeneville by 2½ games and are 2½ games ahead of Johnson City and Elizabethton in the hotly contested Appalachian League West Division. The White Sox are the only team out of playoff contention.
On Saturday, Mets starter Persio Reyes (4-4) got roughed up in the first inning, giving up five runs on seven hits, including a two-run triple by Sox third baseman Trey Michalczewski.
Kingsport (31-24) got on the board in the second when Jeff Diehl led off with a solo homer and Eris Peguero plated another run.
Diehl and Amed Rosario had RBI singles in the third and Jonathan Leroux doubled in a run in the fourth to make it 6-5.
But the Mets could get no closer and Bristol scored three runs in the ninth to put the game out of reach.
Thurman Hall went 4-for-5 to lead the Sox (15-39) at the plate. Victor Velasquez, Toby Thomas and Michalczewski knocked in two runs each.
Jonathan Leroux, Pedro Perez and Amed Rosario recorded three hits apiece for Kingsport, which outhit Bristol 16-14 but stranded 11 runners on base.
The Mets conclude their nine-game homestand with a series against Burlington. Game 1 is Sunday at 6 p.m.