http://www.honkingduck.com/discog/olds_search.php?cs=t&match=smoke+gets+in+your+&ctx=sV0&submit=Literal+Search

This brings up close to 40 78-era recordings, and it's undoubtedly far from complete.

The singer isn't exactly a household name, but this is the earliest recording I can see in the batch:

Selection: Smoke Gets In Your Eyes
Artist: Gertrude Niessen
Date Recorded: 11/01/33
Recording Label: Victor
Catalog Number: vi24454*
Matrix 1: 78377=1

followed closely by
Selection: Smoke Gets In Your Eyes (v Bob Lawrence)
Artist: Paul Whiteman & His Orch
Date Recorded: 11/03/33
Recording Label: Victor
Catalog Number: vi24455*
Matrix 1: 78511=1

There are about a dozen recordings from 11/1933 to 3/1934.