The easiest way to avoid the Southern Strategy would have been for Nixon and some other leaders in the GOP to realize that accepting into the Party of Lincoln people who were leaving the Democratic Party because they were mad about civil rights legislation was probably not such a hot idea long term.
They then state very publicly on numerous occasions that those folks were free to leave the Democratic Party but they were not welcome in Lincoln's Party. Would have been controversial at the time, today they would be hailed as visionaries.
Ri-ight. That's a sure-fire recipe for winning elections: telling large numbers of voters you don't like them and don't want them.
This idea is based on the cartoon version of the "Southern Strategy": that it was an appeal to white Southern racism and included segregationist Dixiecrat officeholders changing parties. Neither is true. Nixon refrained from public denunciations of white Southern racism, but there was no retreat on enforcement of the civil rights laws. The point was to refrain from poking white Southerners in the eye on the issue. The battle was over, and the white supremacists had lost. The old-bull Dixiecrats were still in office as Democrats - and with the exception of Strom Thurmond, they remained Democrats till they died. (There was one obscure U.S. Representative who changed parties in his last term of office.)
Historically, the national Democrats had allied with Southern Democrats to protect white supremacist rule in the South. That broke in 1948. After that, there was no reason for rural, Protestant, socially conservative Southerners to give exclusive support to a party that was becoming predominantly urban, Catholic, and socially liberal. Republicans didn't offer to replace the national Democrats as allies of Jim Crow - it was neither necessary nor possible. All they did was be publicly silent about race issues, and let all the other issues push Southerners to the Republican side. It took two generations - the old-bull Democrats had to die or retire, the older cohorts of "yellow-dog Democrat" voters had to be replaced by younger people who weren't raised from birth as Democrats.
This really did take a long time - as late as 2002, Texas elected more Democrats than Republicans to the U.S. House. And it went top down. Nixon carried every Southern state in 1972 by landslide margins: his lowest margin in the South was 33% in Texas. But Republicans won only 32 of 108 House seats in the region, and didn't even contest 35.
But this process had been coming for a generation by 1968. In 1956 Republicans won 7 of 105 seats in "Confederate" states; they contested only 42. In 1966, they won 23 of 106, contesting 60. Winning 32 in 1972 was a modest step forward; there was no reason for Republicans to spurn these gains.