Tom drank a bit more than his usual share, trying to make up for the unpleasant time he'd had at the hospital.
Literal
Hospital [at-で] unpleasant experience [object-を] doing part [object-を] recovering [intending-と思って] Tom [topic-は] his appropriate-amount [than-より] a-little more alcohol [object-を] drank.
A complex sentence. 不愉快な思いをしている分を取り戻そうと思って chains a goal (取り戻そうと思って 'thinking to make up for it') as motivation. 適量より少し多め uses より for comparison ('a bit more than his usual amount'). 多め adds the suffix め meaning 'on the X side.'