Question about SMC and MagSafe not charging

Hi everyone,

I have a MacBook Air (mid 2013) whose charge has run out. However, when I plug in the MagSafe charger, the computer will not charge (the LED remains dim). I've tried multiple outlets around the house, so I assume the problem is either with the charger or the MacBook itself. I know that many sources suggest resetting the SMC to deal with battery-related issues (press shift, ctrl, option, power). However, does this work even if the MacBook is charge less? When I try resetting, I only receive the default flashing red low battery icon (and the charger continues to not charge).

I have also tried cleaning the MacBook port and warming the charger using body heat, but nothing has worked so far. Is there anything else I can try, or am I doomed until I can get a new charger? (Typing this from phone right now.) Thanks for all of your help.