Liz Cheney would not be elegible due to Wyoming's sore loser law.
WY Stat § 22-5-302 (2014)
22-5-302. Unsuccessful primary candidates precluded.
An unsuccessful candidate for office at a primary election, whose name is printed on any party ballot, may not seek nomination by petition for the same office at the next general election.
It is unlikely that she would want to anyway, her result in the primary puts her in a very weak position. Among Republicans, it is clear that Hageman is preferred by a margin of 2:1. And some of Cheney's republican supporters would not vote for her in the General for fear of allowing the Democratic party candidate to win.
Democrats (a minority in Wyoming) would be unlikely to support Cheney, apart from her Anti-trump stand, most of the rest of her policies are fairly conventional Republican policies (Remember her criticism of environmental groups). And for the same reason, Cheney is very unlikely to switch to the Democratic party, she simply isn't a supporter of the wider Liberal platform of the party.
Independents might support Cheney, but there are not enough to give her the win. Only about 1/8 of voters identify as unaffiliated.
And if she has ambitions to be President, running as a spoiler is not likely to win support among the conservative sector of the American public.