I don’t think UH ever used a block H as a primary logo. From what I can tell, the interlocking UH in some form was basically our first logo, dating back to a club Hockey team we had in the 30s shortly after we became a University. The letter is designed to evoke a letter sweater, which was the way that varsity letters used to be displayed before modern, baseball jacket style “letterman jackets” became popular in the 30s.
@Woodedge compiled a pretty comprehensive history a while back.