Most suppliers set a minimum of 10-25 pieces because setup costs money. Once that cost is already paid, though, there's no real reason a single reorder should be refused.
Screen printing has a genuine per-colour, per-design setup cost that only makes sense to spread across a reasonable batch. Embroidery has a one-off digitising cost for a new logo. Suppliers set minimums to avoid losing money on setup versus what they charge for one garment.
Once your logo is digitised (embroidery) or your screens are made (print, for a fixed design), that setup cost is already paid. A single reorder a year later shouldn't need to clear the same minimum as a first-time setup — but plenty of suppliers apply the minimum regardless, because their system doesn't distinguish "new setup" from "reprint of an existing job."
Ask directly: is the minimum for new logos only, or every order? A supplier who holds your spec on file (garment, colour, logo position) should be able to reorder one piece without re-running setup.
No minimum order — one garment for one new starter is a normal order once your original spec is on file. See what workwear costs for how that affects reorder pricing specifically.
Setup costs for embroidery digitising or print screens only make financial sense spread across a batch of a certain size for a brand-new design.
They shouldn't, in principle — the setup cost is already paid. Whether a supplier actually honours that depends on whether they hold your spec on file and treat a reorder differently from a new setup.
Tell us what you're trying to do and we'll give you a straight answer, not a sales pitch.