Putting on Side
Giving oneself airs. Side is an archaic word for a train or trailing gown; also long, as “his beard was side.” A side-coat means a long trailing coat. (Anglo-Saxon sid, great, wide, long- as sid-feax, long hair.)
“I do not like side frocks for little girls.” —Skinner.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894