The hat as a symbol of Indy's formative moment; the relic hunter gives him a template of how to dress and act, but also defines what he doesn't want to be on an ethical level. It's a talisman of the day he 'grew up' and learned that archaeology isn't always a pure faith, and represents the dichotomy within him between the cynical, grizzled worldview he adopted and the corruption to which that worldliness can lead.
EDIT: It just occurred to me that because Indy wears the hat constantly it may not be seen as a costume change - though I'd argue that that scene in Last Crusade is his single most formative costume change. Also, there is the running trope of him losing the hat and grabbing it back, which I'd argue to signify his refusal to compromise who he is - the films generally have a scene where his identity is questioned or outright forced off him (i.e. Belloq's 'shadowy reflection' speech, or the Thuggee possession ritual),and he's forced to reassert it or take it back - so you could see the hat-grabbing of foreshadowing of this and a reminder that he never lets go of who he is. And I'm pretty sure I'm not totally reaching with that one.
And then we have the Professor Indy costume, which tells us that underneath that boy scout never fully went away. he may be uncomfortable in the life of a professor, but when he's in these clothes there's an honest enthusiasm for learning that tells us he's no brute and informs his romantic centre.
While these two aren't intended to denote any change in the character as we're meant to understand them, they're both put in the right places to understand Indy better.
Edited by Workyticket - 4/25/13 at 9:58pm