The rise and run of the steps could be causing the problem. If your run (the part that you step on) is more than 8 inches, then you could possibly gain more height. It depends on on how many stairs there are. I know this from watching an episode of "Ask This Old House" in which Tom Silva went to a person's home and gave them additional headroom by changing the rise and run. Good luck.

The depth of each step measures at 7-3/4 in. I didn't measure the rise, but would hesitate to make them steeper than they are. Since the 2nd floor landing faces a sloping roof, the stairs probably can't be moved further back.

A bit more on the beam. The beam traverses the width of the house addition, crosses (perpendicular) one side wall of the stairwell and ends over the 2nd stairwell side wall. I'm not sure if this complicates the problem or not.

My intention is to install a beam over the existing one, then, remove the existing beam and install a ceiling approx. 12" higher than current ceiling. If the project requires a lot of carpentry skill, or is dangerous to myself or the house, I'll get a qualified contractor to do it. For now, though, I don't have enough information to evaluate the project.

It's something a professional should look at. We need to see the existing conditions, how to transfer the load from the beam (most likely it will be a LVL or GLM engineered beam) to the post, to the foundation. What load is on the beam to size it properly, what steel will be required, for example strapping to make the continuos lateral loads acroos the gap at the old beam... and quite a bit more info.
It's something that may even require a structural engineer, your local pro will be able to tell from a few test holes to see what's up.
Alter Eagle Construction & Design