The record in the Football League is 8, set by Grimsby Town in a run
starting on 9th April 1908. 2 of the games were (the first 2) in the
following 1908/09 season.

Mansfield Town and Stockport County managed seven, both split over two
seasons, from 5 May 1979 and 15 Apr 1933 respectively.

Several clubs (in addition to Grimsby) have managed 6 in a single
season; the most recent were Spurs from 4 Jan 1987. There were two
home FA Cup ties during Spurs' run, but the overall record was spoiled
by having to play two League Cup games away!

Best wishes,

Tony