For explanation purposes let's use the original flight and date of travel being: 12.25am on August 20th.
The Delta Schedule Change has resulted in your flight now departing at 10.25am on Aug 19.

Your best bet is to call and ask for a change to the next day Aug 20th ... anyway you were scheduled to travel originally at 12.25am on Aug 20th and DL's schedule change has changed your travel to 10.25am Aug 19th ....the day before.
For many travelers that is not a problem ... for some it could be a huge problem as it causes one not to have enough time to complete work that had been planned on Aug 19th.

You should call and ask to be put back on a departure date of Aug 20th, as that was the day planned for your travel. Delta should do this at no additional cost, irrespective of the original fare paid.

Now where this gets complicated is when Delta/ SkyTeam partners do not have a daily service to that city/ airport, they may have a thrice weekly or a five times a week scheduled. Then one may have to be stay extra days just to catch the flight on the 1st possible 'next' day.