Continuity mistake: Columbo stops Mr. Finch on the road after following him from Biltmore hotel; as Columbo is speaking with him, the cigar in Columbo's hand is always in Finch's shot, even after we see Columbo place his hand on the roof. Plus after we see him put his hand down, the cigar ash keeps going from 1" to nothing. (01:44:20)
Continuity mistake: At the end, when Columbo is demonstrating on the mannequin, his cigar changes position between shots.
Answer: In the show, he was already a Lieutenant. A promotion would put him in the next rank up, which for the L.A.P.D. would be Captain. However, some of a Captain's duties would be overseeing other officers and ensuring they're compliant with policies, regulations, and standards. It would also most likely take him out of the field. This is something Columbo has no desire for as he rarely goes to police HQ's. Nor does he show interest in compliance and standards (for example, not going to his semi-annual evaluation at the firing range). However, he could still be assigned to a higher pay grade based on expertise, which is a form of promotion that does not include rank advancement. This would be going from Lieutenant I to Lieutenant II. I don't believe in the show it's ever started what his pay grade is. Although, in s02e01 (I believe) he mentions making $11K a year. Whether or not this was a true statement on his part, if you could find pay scale information for an LAPD Lieutenant in the 70's, it could give you an idea of his pay grade.
Bishop73