Besides the follow up film there also was a follow up book. Walt Disney's Santa's Toy Shop, published in 1950 by Little Golden Books, uses the characters as drawn in the 1932 film in a different story. See more
"Dear Santy Claus, I'm Billy Brown. / Here's a list I've jotted down. / I want a giraffe and a duck and a whale, / A cow and a pig with a seven-foot tail, / A tiger that roars and a leopard with spots, / A barrel of monkeys, a pony who trots, / I want a baboon, a dog that'll bark, / A walrus, a herring, a hog and a shark." Ha! I'll give him Noah's Ark.
But Billy Brown, for seven years, hasn't washed behind his ears.
Well, ha-ha, there's still some hope. I'll just add a cake of ...