For mice nothing is better than the old fashioned snap traps. Bait with salami or pepperoni, stuck into the bait hook on the pan, or if no hook, tie down with dental floss. Set against the wall as willie says, placement is very important the biggest mistake most people make.
But: Mice can't help but leave droppings everywhere they go and are very easy to catch, so combined with the fact that you caught a rat in the crawlspace, I don't think you are chasing a mouse. Norway rats are hard to trap (roof rats not so much). A rat sized snap trap set the same as above for mice may work. I usually just poison rats and accept the fact that I may be looking for a smelly corpse a few weeks later. The trap I have found that actually works the best for them is a chipmunk/squirrel sized live (cage) trap baited with oil type sunflower seeds (bird seed) or dry pet food. Of course, you then have a live rat to deal with...