A second SB would likely get him in the HOF I think.
There's 13 coaches all-time with multiple SBs. Ignore Belichick, who's still active.
Of the other 12, 9 of them have been elected to the HOF.
George Seifert, Mike Shanahan, and Tom Coughlin are the only three coaches with multiple SB rings that aren't in the HOF.
Shanahan and Coughlin very well may end up getting in, because as I said earlier, as history suggests, it can take 10-20 years before these guys get in. Shanahan last coached in 2013, and Coughlin in 2015, so they may have a ways to go. I think Coughlin has a great shot to get in, but Shanahan's may be a bit wider. After the back to back SB wins in Denver, he won one playoff game over the next 14 seasons.
George Seifert is an interesting case. His time in SF was almost unmatched by anybody. In 8 years, he won 77% of his games, had 10 playoff wins, two SB championships, and 5 conference championship appearances. In the grand scheme of things, he didn't coach very long (11 seasons) and he also sucked badly in Carolina for three years at the end of his career.
But moral of the story is... a 2nd SB ring would put John very much in the "heavily favored" conversation to make the HOF.