Hi, and welcome to the board.
As you've just got him I'll assume he's still quite young.
These are my thoughts, but i hope other owners may chip in with their experiences as well.
I've found that young chipmunks are still finding their way in the world, and everything they come across has to be tested to see if it's edible. I found with some of mine, if you poke a finger through the bars for example, they will bite it, just as you describe, with a nibble first and then a harder bite. I guess our fingers look a lot like a nut. They also have a fascination with human toes BTW. It's unlikely a very young chipmunk could break the skin and draw blood though.
If your chipmunk is an adult and you feel if after the exploratory nibble the actual bite is intending you harm, then it could well be fear or a smell on your hands he doesn't like. I've found that nearly all my chipmunks young and old dislike human skin, much preferring to avoid my hands to jump on my sleeve. This is even though I'm very careful to wash off any smell from my hands (scented perfumed soap or cooking ingredients for example).
You've had yours for only a short time, so he probably has not had time to get used to you yet and start bonding with you. Consider how you approach him with your hand. Always let him see your hand coming (avoid surprises and not from above). Move your hand towards him slowly, speaking to him softly. Try not to pick him up or grab him, let him choose to come to you. He's likely to back away at first, but if he moves towards your hand to investigate it, that's good. Perhaps you can hold a monkey nut in your fingers, this should teach him to associate the presence of your hand with something good.
I hope this will put your mind at rest and that something in the above will give you the solution.