Here's how I'd do it:
1) Scuttle the public perception of Nixon's Vietnam policy. Get another major Tet-style PR disaster and Americanization will no longer seem an unqualified success - people might begin clamoring for a faster exit.
2) Thomas Eagleton gets hit by a bus or something in 1971, before he can slander McGovern anonymously to the newspapers. There will be no "acid, amnesty and abortion" rumors, and it will be much easier for working-class Democratic voters to rally behind McGovern. (He favored amnesty, but the other two are twisting the truth - he was pro-life and believed in drug decriminalization, not legalization.)
3) George Wallace dodges a bullet and runs a third-party campaign again.
If McGovern's seen as less of a divisive and radical candidate, he'll be able to convince an establishment Democrat to serve as his VP candidate - maybe Boston Mayor Kevin White, who'd help with the urban working-class Catholic vote. It also means the convention will be wrapped up quickly, and McGovern will be able to make his speech in prime time rather than being delayed until midnight by pissed-off delegates nominating Archie Bunker and Chairman Mao.
So McGovern has everything he didn't in OTL: the conservative vote split by Wallace, working-class Democrats' support, a convention bounce. More support from the Democratic establishment could lead to better campaign managing, which would help fix McGovern's atrocious campaign commercials and could convince him to run on his record as a war hero.
Couple all this with a shakier Vietnam policy from Nixon, and McGovern has a shot.