My take on a height gage with an integral rule holder. I prefer the reference surface on my height gages to face up so I can easily use my NIST calibrated thumb nail to check when the gage corresponds with the height of the object measured.
A more important feature of this design is the rule holder. A collar can be slid up under the measuring bar and locked in place with the projecting SHCS (someday I'll make a proper thumbscrew). Then the bar lock screw can be unlocked and the bar slid around to lie against the scale and allow the height to be read, or, reversing the procedure, be set.
This was built a long time ago so it employs the more conventional fine adjust mechanism found on beam compasses, etc. Were I to build it today I would use a differential screw design which I much prefer in this application.