After one drink, he drank another, it means that he fluidly drank a lot, until he fell asleep
he drank a lot until he got uh... sleepy?
In the given context, "drink after drink" means that the detective consumed multiple drinks one after another, without a break or interruption. It suggests that the detective continuously drank a series of alcoholic beverages, possibly to excess. This phrase emphasizes the detective's excessive and continuous drinking behavior.
