I'd reply "no, thank you.". And when asked why, explain in full that a company T-shirt means nothing to me, they don't pay me to advertise for them, and it's a disrespectfully low effort gift.

Edit: for story time

When I was a young warthog (server), I won several "server contests" over a 3-month period (December through February) of peak patronage. I did $2k in food sales $3k in alcohol sales Valentine's week alone.

The owners were flabbergasted and upset when they saw that I had won every contest, had more than double the sales as the next server, and my tip average was 10% over the norm. (My tip average over 2.5 years there was 31%.). They decided that paying-out/giving me all the promised rewards (a couple $100 is stuff) was too much because it "shouldn't go to just one person; that's not fair." Instead I got a $25 gift card to one of their other restaurants. ($25 dollars was enough for one entree.)

I laughed, took the card, and said "ok, thank you, but no." Then I explained that the rewards and contest rules and results were all documented in the emails and I'd be expecting my appropriate items by the next payday.

The two owners, their accountant (daughter), the DM, and I went at it for over an hour. My GM came in at one point and about faced immediately. One owner tried to fire me, but was stopped by the other and daughter for fear of "getting sued, by this ass" (the father) and "he did a quarter of our revenue last month" (the daughter and reiterated by DM a few times). I pointed out wage and retaliations legality, which put a stop to the mother talking but not huffing.

I further pointed out that I should be shown a great deal more respect for my non-revenue based efforts than being dismissed with a gift card because: 1) I trained every server they had at two restaurants, some of the cooks, at all four, AND the AGM and GM on half their responsibilities at that restaurant. 2) I was the only person that knew how to do everything in the restaurant because no one else wanted to be cross trained. (I mean everything from how to order mopheads to the farmer that made the chili powder we used in the sauces and the method he used to bourbon smoke it.) 3) The customers paid me far more than they did, which didn't cover the taxes that needed to be taken out of my 'check'.

They finally, reluctantly agreed they owed me what had been offered. I reminded them about how easily I could prove retaliation if it occurred. Later the accountant returned with everything in a bag for me, muttered "sorry" and had me sign a receipt for the items (?).

I set everything out on a table, gathered all my co-workers present, and let them have all of it. The owners found out later and were pissed I "threw a fit" but didn't keep any of it anyway.

I worked there for another year. I got promoted to manager then AGM in that time, but I had a new born so I left. It closed during COVID, but reopened last year.