Verb for increasing the vertical dimension of a space?

Heighten

v. to increase the height of; make higher.

From the Farlex Free Dictionary

I couldn't come up with one word, and I know you said in your question that raise is not the right word, but it looks like when engineers enlarge a tunnel by making it taller they say they are going to raise the roof. They also use the phrase increased clearance for a tunnel that is made taller.

(And I know you didn't ask this, but it also appears that tunnel roof is preferred to tunnel ceiling.)

See this link for use of "raise the roof" in more technical documents.

See this link for NGram of tunnel ceiling vs. tunnel roof.

The only option seems to be to expand vertically, unless there's a surprise candidate!