Yes, send it. You do not need the tier or housing unit. The facility's mailroom knows exactly where every inmate is housed by their inmate ID number, and that is how mail gets routed internally. The housing unit information is helpful but not required on the envelope.
All you need on the address is the inmate's full legal name, their inmate ID number, and the facility's mailing address. The mailroom does the rest. They sort incoming mail by inmate number every day and know where to deliver it without you having to specify the tier.
Address the envelope like this: Inmate Full Name, Inmate ID Number, Facility Name, Facility Street Address, City, State, ZIP. That is sufficient for the letter to reach them.