Here's my experience today (for a flight tomorrow) and some about 4 weeks back mixed in:

• Booked refundable first class in I on outbound
• Wanted to SDC nonstop->nonstop

1. Android App: Same day change and change flight both lead to the same screen, asking $1300 for the listed option
2. iPhone App: Same day change and change flight both lead to the same screen, presenting no option
3. Online Check In at delta.com (click check in, paste/type confirmation and departure, push go)
   1. Had to click more flight options as only standby was there, but the option appeared on the next page
      
4. Using the "Change Flight" option on delta.com under my trips
   1. 
      
   2. 
      
   3. This will only show flights departing up to the current time on the current day. Changing the day to +/- seems to bring up repricing. So SDC to a later flight is only possible via this method by waiting.

I didn't go through with a change this time, but 4 weeks ago I did something similar using the delta.com change flight option from my trips:

• Tried to change via text message and was told "Sorry you have a Travel Agent ticket" (concur) and they refused to go any further with the SDC
• App gave no options
• OLCI did not show the flight I wanted
• Change from mytrips did show the flight and upfared it to J, though it did not re-issue the ticket that I could see; no issue flying it
• Change trips did show a connection option, but it wanted a re-price from the non-stop;
  • When I checked outbound with a connection, it was not asking for any additional charge but changing to non-stop was a re-price

Obviously YMMV but giving some info on what worked for me recently.