How can a soldier reclass before their reenlistment window is open?

He should be able to do a DA 4187 requesting an MOS change from his commander.
If approved, then it will be a matter of finding an MOS that he qualifies for and has available billets for training.
There may be consequences to his contract, though, so a thorough review of obligations and stipulations in his contract should predate any actions going up for approval.

Voluntary reclasses are handled through your Career Counselor. There are a lot of factors involved starting with the class seats available for reclass, time on station rules, and MOS strengths. Then the counselor has to cross reference to verify other factors such as, time in grade, rank requirements, clearance, line scores, and a slew of other things. Your soldier won't be eligible to reclass into anything at all unless they have been on station at least one year.

You should also take a look at the most recent IN/OUT calls to see what may be available. He'd only be able to reclass into something with a shortage. Some MOSes like 17C can be applied for at any time, but require a full packet of information.
https://www.hrc.army.mil/Milper/17-296