For a guide to the 4 hole harmonica, I'd recommend you check out The Ultimate Miniature Harmonica Tunebook by Pat Missin. I haven't read it, but the author is very respected in the harmonica world.
To answer your question: Yes! You can play all the 13 chromatic notes from your low C to your high C. (This might even be possible without overblows!) If you really push it the range can maybe be extended, but not by very much, and you might have to take your instrument apart to set it up.
The easiest extra notes are produced by draw bends: C# in hole 1 and G# in hole 3. Following some online guide to draw bends (there's probably hundreds of them) they aren't that hard to make. Quite a bit harder are the notes Eb, F# and Bb, which can all be played with something called overblows. It's not easy, but with some dedication it's possible.I never heard of anyone playing overblows on a 4-holer before, you'd be the first :) You might very well have to get someone to help you get the reeds adjusted to make the overblows work, especially if your instrument is on the cheaper side as most (all?) 4 hole harmonicas are.
Another approach entirely would be to do something called half-valving. This means installing small valves over your draw reeds and some of your blow reeds. The benefit is that blow notes can then be bent down in pitch, allowing you to play all the chromatic notes without overblows. This would also let you bend your low C down at least to a B, maybe to a Bb. The drawback is that you get a more complicated instrument to maintain. (and you would not be able to do overblows)
Installing valves is a bit complicated, but some people find this easier than learning overblows. I personally prefer the overblow approach, and would recommend it over using valves if you're motivated to learn, but others might dissagre.
As for the range, there's one more way to extend it:
There is a technique even less used than the overblow, namely the overdraw. The sound could be rather shrill for such a high note and short reed, but if you can learn to overdraw you would be able to play a high C# above your high C. Maybe even somewhat higher notes. What I said about adjusting your reeds applies even more for this note.
If you're looking for an instrument of higher quality, which could help in learning advanced techniques, Seydel have a 6 hole harmonica that might be small enough for your needs. For a reasonable fee they will custom build it with any layout you want, so you can get four of the holes the notes following the same pattern as your 4-holer. That way you could go lower, and you get an instrument with the same quality as a good blues harp.