The Indy Hat Roll is more of a "last minute grab of something/someone before it is too late" but Indy's Hat itself (and similar objects you mention like it) is probably more an example of a Number One Dime:
The Number One Dime is an item that is in and of itself of little to no interest or usefulness (i.e. it is not a usable weapon), but to the character, it is ridiculously overimportant in a strictly sentimental way.
Indy earns his hat as a child in the introduction to The Last Crusade, when he tries but fails to save the Cross of Coronado from some graverobbers. The lead robber gives him his hat, with the quote "You lost today kid, but it doesn't mean you have to like it..." Thus, the hat is a symbol of Indy's resolve to win, even in the face of overwhelmingly bad odds or previous losses.
Overall, his hat is a Nice Hat:
A discerning hero/villain just can't be seen without their Nice Hat! Harming the nice hat may be akin to kicking the dog, with disastrous results.
To show that you mean business, it's important to have style. Enter: the nice hat. It may actually have utility — even conveying special powers — but more often than not it just looks cool, or even just sexy.
In fact, Indy's hat is called out in the film sub-page for Nice Hat:
Indiana Jones' spiffy fedora, which he never leaves behind even when he's about to be crushed by slowly descending doors. Many fans know it simply as "The Hat".