Cats just break your goddamn heart.
About 20 years ago, I found a helpless, starving little kitten that fit in the palm of my hand, and I took it in and raised it, and I let him come and go as he pleased. I brought him with me a couple of years later when I moved to Cali, and after another year he disappeared, and it tore me apart. I swore I'd never again get attached to another cat, but within a couple more years, another starving little kitten came up to my doorway, so I fed him before I left for work. when I returned over 9 hours later, there he was waiting for me, and so the story repeated itself. A couple years later, another heartbreak when he never returned one day.
About 10 years ago, I moved to the California high desert, and recently one of my neighbor's barn cats had been coming around. I wanted to let him know he was welcome there (I had an ulterior motive - I wanted him to kill any mice that might be near my house), so I fed him a couple of times. He always kept his distance, as he never fully trusted me. Well, a couple of Saturdays ago, I left one of my sheds open all day, but I forgot to check if he was in there when I closed it up that night. Friday morning, 5 1/2 days later, I was getting ready for work and I see his little head pop up in the window of the shed. I got him out of the shed right away and he looked to be OK, so perhaps he feasted on mice and bugs while he was in there. I immediately fed him some premium albacore canned tuna because I felt so guilty. I saw him the next day and fed him some chicken, and the next day some barbeque pork ribs. The day after that he started rubbing up against me wanting more food, and now I can't get rid of him!
I figure I'll keep him fed just enough to hang around and keep the mice away, but I'm going to keep my distance from him because I don't want another heart break.
It's my sincerest hope that the OP gets his cat back soon. It never occurred to me that cats could go missing because they get trapped in sheds until I accidentally trapped one in mine.
I agree with Salsprocket and 02gf74; be proactive. Knock on neighbors doors, explain the situation to them , and ask if they can check their sheds, basements, etc. when they get a chance (They surely don't want you snooping in there, but they might not have reason to open them for a long time unless you ask them to do it on your behalf). Be sure to leave your contact info with every one of them. Start with immediate neighbors and then eventually to every neighbor within a few blocks radius from your house (the distance a cat would typically wander from home).
Best of luck to you.
Life! Life, do you hear me?! Give my creation LI-I-IFE!