The easiest path here is for the South to remain competitive, which isn't quite as far-fetched as one might think. In some states in the deep South, black voters are nearly 30% of the electorate. For a Democrat to win, they'd need to pick up less than half of the 70% that constitutes the white electorate, about 29% if the black vote is 100% Democratic. But it isn't, so let's say Democrats need a third of the Southern white electorate to remain competitive. I'm going back to the 1980s/90s here, so I'm assuming away the latter day rapid growth in the Hispanic population.
There are any number of ways to get there: increased Northern migration that boosts the Democratic vote, less strife in the 60s-90s "culture wars", no Reagan realigning reelection in 1980, a more accepted Civil Rights movement with less violence. I should think getting rid of Vietnam might help things along as well. Civil Rights and the '60s/Vietnam broke the back of Democratic party ties among white Southerners that went back for more than a century.
In any case, a more competitive South means a more competitive rest of the country as the Republicans are forced to moderate their views to appeal to a broader part of the country. A more moderate GOP might remain competitive in places like New Jersey and Connecticut and perhaps even California, which the GOP actually won in 1988.
Now, I'm not saying any of this is particularly likely, but these are, I think, the general contours of the kind of political climate you'd need to increase the number of competitive states.