P.S.
Maybe there is some non-restrict rule: "пода(ва)ть" is preferable if object (slightly) goes up and down during process of transfer. For example "подать нищему (милостыню)" (get coins and drop them), "подать мяч" (in the game), "подать документ" (get it up from one table and put down to another table), "подать руку" (hand goes up for handshake or goes down if you help somebody to get up from ground). Maybe it makes sense. But it doesn't cover all cases.