Look at smaller rescues, they usually have dogs in foster homes rather than kennels. Therefore, dogs already in family environment where they can be assessed with people/children/cats/other dogs etc.
Smaller rescues often more flexible regarding adoption with younger children. Larger rescues very risk-averse and afraid of bad PR headlines.
Once you've found a few rescues you like, get homechecked by them. Then you'll be a position to rehome quickly if a suitable dog comes up.
Join the FB groups related to all the rescues, they are usually much more up to date than websites. FB groups will usually feature dogs - the more popular, "pretty", cute who will be snapped up before even getting to be on the website. So you have a better chance if you're on them.
On the FB groups, you can post comments/reminders that you are looking, so rescue doesn't forget you! Smaller rescues run on a shoestring, time poor, very few people so you can't expect them to remember and ring you each time a (potentially) suitable dog comes in. It's up to you to keep looking, being in contact.
Where are you located? I'm sure people here will be able to give you names of good rescues in your area if you can say North, South, East or West!