To echo the above poster, no, Delta will not pay for your rental car, and your options are the following:
(1) Fly the new time and ask your ride to wait, or pay for your own rental car or taxi
(2) Call Delta and ask for an earlier flight
(3) Call Delta and ask for a refund and book another airline.

Delta happens to be a lot more flexible than most airlines in this regard, in that even with 15 minutes they probably will allow you to change to an earlier routing - and you should be able to actually do that online. No airline will pay for "amenities" such as rental cars, taxis, or hotels incurred as a result of a schedule change announced in advance.