What if he worked for the Soviets?
If the thirties is his mythic resting place, and he's placed into the fifties, then the character is likely to become more reflective of his age and in turn, death, since he's lost his era.
If the thrities are a metaphor for the Old West; and if the fifties are figuratively the New West, then Indy would be the modern equivalent of an cowboy, and in this case, an subscriber to a earlier way of thinking; he's anachronistic.
But what if he became suspicious of his own government? What if his social grievances around the world has blacklisted him? And this is not too unlikely in the McCarthy era; keeping a low profile on someone else's soil, surely Indy would seek refuge somewhere abroad, much like an outlaw would. Maybe for the sake of the story he's on a payroll with the Soviets to seek artifacts for ulterior motives, and Indy realizes this but isn't morally obliged to do anything, because he could care less in the big scheme of things until...
He realizes what the Soviets real intentions are, and then takes matters in his own hands to stop them...world domination, that sort of thing.
But overall, he's become an ex-patriot and he's strictly a independent.