It's been a while since I last saw Origins, but even then I could never get the exact timeline straight. All I remember is that Stryker first recruited them during or immediately after Vietnam. So unless I'm forgetting something this is either while he's still with the team, or just after he quits.
By-the-by, another reason why Logan in the better choice as the Sam Beckett of this movie: this is a way for him to finally regain his old memories. I mean if he's "leaping" into his younger bone-claw era body then it should have an undamaged brain and all of his old life memories.
Mind you, didn't he already remember some of it? In 'The Wolverine' he seemed to remember saving that bloke he saved at Nagasaki (or was it Hiroshima?)