Dynamic range of an image (as opposed to an image sensor) is simply the ratio between the brightest and darkest pixel values. An image of fixed bit depth thus has a fixed maximum dynamic range.
Increasing contrast will increase the dynamic range but only up to the maximum possible given the bit depth.
Whether or not increasing contrast will improve an image depends on many factors. More contrast will make an image bolder, more striking, but too much contrast can destroy an image and leave it looking distinctly "unreal". A low contrast image will invoke different feelings and there are times when this is more appropriate.
Here's an image which I decided looked better with a lower overall contrast:
First rough edit.
Final version. [1]: https://web.archive.org/web/20150907230906/http://mattgrum.com/photo_se/dan_contrasty2.jpg [2]: https://web.archive.org/web/20150908101127/http://www.mattgrum.com/photo_se/dan_other.jpg