It would actually be best to work with another pidgey, as that’s likely what you’re most used to, and its disposition and skillset will closely match your original emotional support pokémon’s. Sticking with the species you’re most used to will help maintain a sense of security and normalcy, which you may need at moments when your mental health is at its lowest. Just remember to give the next one you get an everstone. You may want to specifically ask for a pidgey that doesn’t wish to evolve to avoid stressing your new emotional support pokémon out.
If you wish to try something new, though, there are plenty of options for pokémon that can’t evolve (or can’t do so on their own) but fit those criteria. Munna, for example, is a rather popular choice, but if you wish to stick to flying-types, togetic is small, can’t evolve unless you give it the extremely rare shiny stone, and is well-known for being gentle, caring, and fantastically good at sensing the emotions of its trainers. Woobat, being part psychic, is also fairly popular, and it doesn’t get much bigger as a swoobat. As for popular emotional support pokémon that can’t evolve at all, you may wish to consider farfetch’d, delibird, oricorio, or chatot. Delibird especially is noted for being fairly easy to train as a support pokémon, partly due to its kind nature and mostly due to the fact that it’s not particularly well-suited for battle (meaning if it interacts with humans, it’s often to serve in a capacity other than on the battlefield … which in turn means it may already understand that its relationship with you may be more domestic in nature).
My advice would be to work with your local pokémon shelter. Don’t be afraid to ask questions and inquire about which pokémon would best suit the role of a companion to you, rather than as a member of a battle-ready team. Most pokémon shelters are run out of centers, so the staff would not only completely understand if you’re seeking an adoptee for medical reasons but also be prepared to help you if you have any questions.
But above all, though, remember that the best emotional support pokémon is actually the one you feel the most comfortable with. So while you’re at the shelter, you may wish to keep your mind and options open. If you see a pokémon that you immediately feel drawn to, consider that one. (Of course, ask the staff if the pokémon that calls out to you has a calm disposition, but other than that, you should be fine.)
Best of luck!