As long as you check them frequently (every 2-3 hours at most), good live traps are great. Here's my favorite: https://www.amazon.com/Tomcat-Single-Catch-Live-Mouse/dp/B012RGN4G6/ These can't harm the mouse (they rely on gravity to close, there are no springs). The one downside is that they are opaque, so the only way to tell if there's a mouse inside a closed trap is by weight, motion, or when you open it (less of a problem for tame mice vs wild ones). I also put a piece of light colored tape on the top of the "door" - since the entire trap is black, this makes it easier to see if the trap has closed (meaning you have hopefully caught your mouse).
In the meantime, if you have a 2nd tank/cage, place it on the floor - put some of the bedding from the original cage into the 2nd one (so the cage smells famliar). Also include a water bottle/dish and some food, additional bedding and an exercise wheel. Leave the top open, and place books, boxes, etc. around the cage so the mouse can easily climb into the cage. I have caught several escaped because I heard them running in the wheel in the open cage!
Below are two very common live traps that work well for baby mice (many of the larger traps aren't sensitive enough to be triggered by a baby's weight).
Tomcat: https://www.amazon.com/Tomcat-Single-Catch-Live-Mouse/dp/B012RGN4G6/
I've had the most success with the Tomcat, but no single style is good for all mice. I recommend placing a piece of light colored taped on the front of the door. This makes it easier to tell from a distance if the door has closed. Because the Tomcat is completely opaque, it's hard to tell if there is a mouse inside (especially with a baby mouse), so it the door has been shut, assume there is one (to make sure it doesn't escape again), but don't be surprised if there's no mouse inside (a mouse will sometimes "trip" the trap by running on top of it). Also (especially with baby mice), they can sometimes remove the food inside without setting off the trap. So even if the door is open, check it periodically to make sure there's still food inside.
Victor: https://www.amazon.com/Victor-Multiple-Catch-Humane-M333/dp/B004CMNWES/
You can't tell from a distance if you've caught a mouse with this trap, but the Victor is translucent so you can hold it up to a bright light to see if there's a mouse inside. However, there are some "blind spots" where baby mice can hide from this type of inspection, so even if you can't see a mouse inside by holding it up to the light, open it up be certain.