Most likely yes. Facilities generally make every effort to deliver mail to the correct inmate even when the inmate number is missing from the envelope.
Mail is treated seriously inside correctional facilities. Prison administration broadly understands that family communication is one of the most important factors in successful reentry and rehabilitation. A letter addressed with the correct name and facility that is missing only the inmate number will typically be matched to the right person by mailroom staff rather than returned or discarded.
That said having the inmate number on future correspondence is always better than not having it. The number ensures your letter reaches the right person quickly without requiring manual matching by staff. At larger facilities housing hundreds or even thousands of inmates, a common name without an ID number creates more room for delay or error than a smaller facility would.
For Dyer County Jail in Dyersburg Tennessee specifically, if you need help locating his inmate number InmateAid's free inmate search can help you find it using his full name and date of birth. Having that number on every piece of mail going forward takes the uncertainty out of the delivery process entirely.
The letter you already sent has a good chance of reaching him. For everything you send next make sure the number is included.
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