I think you have to sail under an unlimited HP chief to get unlimited time as first to apply the time toward a chiefs unlimited. I was hired in the GOM for that reason as both regular chiefs had limited HP chiefs/unlimited first license and needed 6 months sea time as “first engineer” under an unlimited chief.
“What do I know, I’ m just a humble marine engineer.” (Dirk Pitt in Clive Cusslers book Cyclops)