Flatter character designs were in animated cartoons everywhere by the mid-‘50s. Here are some examples from Woodpecker From Mars, a Walter Lantz cartoon released in 1956 and directed by Paul J. Smith.
The Lantz cost-cutting is evident, too. All the characters are stationary except for the heads of the ones in a straight-forward view along the street. Lantz also reuses animation of the Walter Winchell caricature from Termites From Mars.