At the start of the 3rd Test of the 1938-9 series in South Africa, the England batting averages were:
Gibb 85.66
Valentine 77.60
Hutton 61.70
Hammond 60.75
After the match, Hutton's average had dropped to 58.90, but Paynter's had risen to 67.00 (and the other 3 were still averaging over 60). Gibb's average dropped below 60 in the next match.
In the 1st Test against WI the following summer (with Valentine - and Gibb - no longer in the team), Hutton's average went back above 60 to join Hammond and Paynter.
A near miss: Headley, Weekes and Worrell all averaged 60+ at the start of the 1953-4 series against England, but Worrell missed the 1st Test, the only one which Headley played in.
Another near miss: Sutcliffe and Hobbs averaged 60+ before the 3rd Test in Australia in 1928-9; Sutcliffe and Hammond averaged 60+ after it, but Hobbs's average had dropped to 59.56. (Technically, Duckworth's average was infinite before the 3rd Test, but as he'd only scored 53 runs in his 4 not-out innings, that doesn't really count).