Delta recently changed my flight (yesterday). It's not a huge change, but it will be a later arrival time than my original booking. If I miss my ride to a wedding because of it, will I be able to get Delta to pay for my rental car, even over several days?

I read that airlines are flexible in this situation, in that they'll offer a different flight without a change fee or fare difference payment and/or flight cancellation (full refund). I'm wondering, though, if they offer different amenities for my situation.

Thanks in advance!