The answer is retailing. Milady defines it in nearly the same words the question uses: retailing is the act of recommending and selling products to your clients for at-home use.
The book pairs retailing with upselling as the two main techniques for increasing income, but treats them as distinct: upselling typically means adding a service or upgrade during the visit, while retailing specifically means selling take-home product. Ticket upgrading and servicing aren't defined terms in this section at all.
Milady Standard Barbering, 6th ed., Ch. 20 (Working Behind the Chair), p. 746: "Retailing is the act of recommending and selling products to your clients for at-home use."
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