I volunteer with our humane society, though we ended up adopting from another rescue. Our dog was being fostered in a busy home with kids and other pets... like our house. When I saw her and her description it seemed like a perfect match! Our girl (a poodle mix mutt) is totally happy in a crowd or chilling on the couch with us. She’s super smart, friendly with other dogs, and LOVES everyone she meets.
All humane societies are independently operated. Ours sound similar to yours though in that most small breed puppies/young dogs get adopted very fast- the younger the faster they go! That means the staff/volunteers don’t get to know the dog that well beyond basic temperament testing. The one exception is if the dog spends time with a foster before being placed up for adoption. This happens with puppies that still need to be with their mamas, or who need time to heal or get some training.
Sometimes if you come in a few times and fill out the paperwork to be in the system as an approved adopter, the counselors will give you a heads up if they see an animal that’s a good match for your family. They may know what puppies and dogs are currently in foster care and will be available for adoption soon. You might call and ask about this at your local shelter. Right now they probably have more animals than usual in foster care. This is a tough time for rescues.