While I am but a novice when it comes to impact evaluation, I would say whether your comparison makes sense depends on whether it satisfies the parallel trend assumption.
The whole point of dif-in-dif strategy builds on this underlying assumption of similar trajectory of evolution in the absence of any exogenous shock. Ideally, in your case that would be all three period altogether since you want to observe the pre-treatment, treatment, and post-treatment period, each of which reveals different information with respect to the behaviors of the two groups. As long as this assumption holds, and there is no other time-varying factors that affect both groups differently, especially during the treatment period, the difference between period 1 & 2 reveals the impact of the treatment. The difference between period 2 & 3 is nonsensical, however as the treatment and control groups are already different in the 2nd period. If you want to compare the immediate effect and the long-run effect of the treatment then I think you should compare the difference between period 1 & 2 and the difference between period 1 & 3. Be mindful, however, of other time-varying factors that could affect the results other than the treatment in the long-run.