I have never heard the term raising a mustache other than when Ward says it to Wally. Have you?

No, I've never heard that one either. I've always heard, growing a mustache or beard.

Maybe it's what they said in the Seabees.

It's Shaker Heights slang.

It was an old expression back in the 1800’s and early 1900’s. In Sister Carrie Eddie Albert says he might try raising a mustache.