He voted to allow a STAY of execution to continue until it expires. When it does, Alito will let the execution go through.
I don't think that's what's going on.
I believe Taylor's (cruel and unusual based) appeal is at the 8th Circuit Court of Appeals which will hear arguments.
If they grant or deny the appeal, Missouri or Taylor gets to appeal and the hysterical folks just might get what they want--an 8th Amendment case where Alito gets to please or horrify us with an actual vote and opinion on the matter.
Two other similar cases are in Florida, both of which were stayed by the SC.
Of course, I could be wrong.