Air side transit is possible at the transfer desk for full service airlines and certain LCC to other airlines they have interline agreements with, or you have both flights on the same ticket.

There is no interline agreement between Delta and Indigo. But you have no check-in luggage so a slight chance that air side transfer desk may be able to issue boarding pass.

"Please proceed to your nearest transfer lounge/counter. At the transfer lounge, you may refer to the Transfer Information Passenger Display System (TIPDS) to find out which transfer counter you should approach. The check-in agent at the counter will process your check-in for your departure flight."

Transiting Guide | Singapore Changi Airport