Technically, according to Pope Pius XII, a priest is also supposed to wear his surplice at Mass, or else he is not properly vested.
How many priests do you know that wear their surplice at Mass???
Look it up!
Surely not an alb, surplice and chasuble? Is there a vesting prayer for a suplice in the prayers befor Mass?
No.
The a surplice is just a shorter alb; the two are interchangeable, technically, for vesting purposes.
Here, you can see SSPX deacons at an ordination Mass wearing the surplice under the dalmatic, rather than the alb:
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