Mice are VERY skittish critters. I had mice, rats, hamsters, and guinea pigs growing up and all though they are all in the rodent family, they all act differently in one way or another.
Mice are hard to "tame". They will always be skittish to some extent. But like the others said, take it slow and take your time. It will take a few weeks for your mouse to get used to you. Its mostly movement and smell that scare them, noise can too, but not so much. Cutting up a shirt is a really good idea (I never thought of that!) so I would suggest doing what she said about it, and the mouse will most likely chew on the shirt, but it shouldnt hurt it because, it is a mouse lol in the wild they chew through rubbermaid storage tubs in peoples sheds (like ours) and insulation with fiber glass and much worse things than a bit of shirt, so it should hurt it but if you notice it chewing on it, just slowly wave your hand where it can see it near the cage and it should get distracted and stop (who knows, maybe it will associate the smell of the clothing with the waving of the hand and it might help with the skittishness). But yeah, when reaching into the cage, just be verrrry slow, you can make little noises like whispering, or ticking noises like you would make for a cat but quieter, etc. that may sooth it a little bit.
Dont know what to tell you, when we got our mice, we got to hand pick them and we picked the most mellow ones in the cage so we never had problems with them really, the guineas we had were more skittish than our mice -shrugs-
Good luck with it and let us know how it turns out!