Nothing. If you're desperate, palmer and similar words slant rhyme.
==If one allows proper names, there is "The Short Happy Life of Francis Macomber," a short story by Ernest Hemingway.
Macomber and Mcomber are unusual names and perhaps should not disturb the finding that no English word rhymes with somber.
Of course, Ira Gerschwin and Ogden Nash would have found rhymes that no one else could have dreamed of.
Some words that rhyme with sombre are hombre, somber, and cumbre.
sombre, décombre, encombre (from verb encombrer)
A funeral is a sombre occasion. There was a sombre atmosphere at the cathedral.
Sombre was created in 1998.
The duration of Sombre is 1.87 hours.
The word sombre, meaning "depressing" is an adjective.
Sombre Pigeon was created in 1876.
A Sombre Dance was created in 1999.
Sombre catshark was created in 2008.
Sombre Kingfisher was created in 1850.
Sombre Reptiles was created in 1975.
Sombre Romantic was created in 2001-02.
The population of Lafage-sur-Sombre is 119.