We found a garter in our home 2 winters ago. We caught him around February - so spring was just around the corner. I knew nothing about snakes at that time, other than they fascinated me. Found a large, very high box that we put him in until we could release him. I bought a pinky - he wanted nothing to do with it. Did some research, and found out garters eat minnows, frogs, and night crawlers. Bought some minnows, and put them in a small tank in the enclosure. The garter actually went into the tank with the fish (had a separate water dish), but never ate the minnows. I went and bought a dozen earthworms (aka night crawlers) and the garter loved those! Gobbled them up pretty quickly. Feeding just worms is not a balanced diet for a garter apparently, but since he was just a guest for a couple of months, I was just glad he was eating something! He was released that same spring, and I swear I see him outside on our back patio, in our back garden. Might not be him, but... (we have lots of snakes on our property).
Try some earthworms. Worked for us. Hope this helps! Good luck with the little guy.