Everyone who loves their pet and has run out of options for re-homing would love to be able to place them in a restricted access shelter, where they can be confident the pet will have a home until it's new forever home can be found. Unfortunately this is not always possible. A restricted access shelter in our area reports it has space for 250 animals and gets 300 requests for placement per week. Any number of factor can prevent your pet from being accepted at the shelter, it may or may not be possible to influence placement
Local advertisement placed so you can re-home your pet directly, may not bring results. Even if people do respond, they may not be acceptable.
If your pet is older, finding a good home may be extremely problematic. Two options to consider are long term boarding, or euthanasia both of these options have significant implications and would be your very last options.
If your pet is a papered pure breed, or on looks like they are a pure breed, contacting organizations that specialize in that breed/species may lead you to list of people interested in your pet, but don't have the funds to purchase.
Use your personal and professional contacts to search for people considering a pet. A large percentage of re-homes are friend to friend, of friend of a friend.
Lastly don't rule out open door shelters, open door may not mean what you think it does. If you do take your pet to an open door shelter you can improve your pets adopt-ability by include a bio sheet that talks about their good qualities, and states why you had to be separated from them. If there are several open door shelters in your area, visit them, see what they look like, the ones with lots of volunteers are going to be the safest. Ask specifically about whatever concerns you have. Some shelters will include the intake date on the pets description (I know ours does). If you see intake dates that are week and months ago, and you see volunteers taking dogs for walks, cuddling cats, and taking bunnies for romps you can sure that everything possible is being done to ensure positive re-homing occurs.