It is quite clearly not an, It would be
Let's make a hire.
Unless we speak different languages of course, which in that case I still don't think it's correct.
An is used if the first sound coming out of it is a vowel.
A is used if the first sound coming out is a consonant.
Which in our case issss H!
That is a consonant.
Meaning, no you do not say let's make an hire.
It is let's make a hire.
@Karoku