And in 1957, here are the appearances by aliens in Batman stories:
None. Zip, zero, rien. In Detective #250, an alien rocket does make it to Earth, giving a crook some fantastic weapons, but the aliens themselves do not appear.
In 1956, the same thing. Oh, Superman does have a cameo in Batman #101, but that's it.
So graphing 1956-1966:
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A big bulge in 1961, but pretty sharp declines after that, so it looks like Schiff was phasing out the aliens before Schwartz took over.
I am aware of at least two other Batman and the Aliens stories; in Batman #41 (June-July 1947), Batman battles a Martian villain, and in Batman #78 (Sept 1953), he encounters The Manhunter from Mars.
Incidentally, the alien craze of the late 1950s-early 1960s was not solely limited to Batman and Jack Schiff. While doing some research on the Archie series, I came across this cover:
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That's also 1961.