What is the origin of the phrase "Put one in the wood for him"?

It's usually put one in the barrel...

I asked the landlord for a pint, pointing at Duncan. He pointed at two full ones lined up at Duncan's elbow but agreed to put one in the barrel for him.

Admittedly it's a slightly odd usage, in that the pint of beer is already in the barrel, so reserve might be a better verb than put. I'm from Sussex, and I note OP is in Kent, so perhaps the pint / wood / barrel form is essentially a South East UK usage (but it's probably more widespread).

I don't think there's any reason to look to an "origin" as such. If your friend arrives at the pub after you, you might say "There's a pint in the barrel" (already paid for by you). Americans don't drink pints, but I'm sure if one was eating a slice of pie in a diner he could say "There's a slice behind the counter [for you]" when his friend arrived, and expect to be understood.

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Here's a link to sociolinguist Kate Fox in Watching the English (2005) talking about it.

"Put one in the wood" is a competitive dart throwing term. The bullseye is made of wood while the outer board target regions are made of other material such as horsehair.

I think "one in the wood" is a Kent/ south east thing.

In Kent you can say I'll have a Stella and "one in the wood" and be understood no problem. Say it elsewhere and you'll get blank stares. But it means you got one behind bar ready when you are. (Bar=wood).

I used to work as a barman when I was around 20 and used to drink in the same pub and used to rack up loads of pints in the wood from people buying drinks for the staff. Got 18 pints in the wood.