The issue I see with your approach is that you will not be able to see anything about the pre-treatment differences unless you have very precise information about the experiment or policy. It will be hard or even impossible to say something about the common trend assumption between the treatment and control groups which is a vital part of difference in differences.

For instance, say you have a job market program which is mandatory but in period 1 only motivated individuals will attend it. In period 2, which is the starting point of your data, the policy maker forces the other individuals to attend the job market program, and finally in period 3 you see all "treated" individuals. In this case it is hard to claim that those treated in period 1 and those in treated in period 2 have the same trend in their outcomes

1. due to the unobserved factors that led to treatment selection in the first round
2. due to the fact that individuals in period 1 have already been treated so their trend already changed (if the policy had an effect).

Of course this is a very artificial example and problematic mostly because treatment is non-random but I guess you will see the point. Without more knowledge about the experiment you can not credibly sell a difference in differences analysis in this set-up because you cannot say anything about the pre-treatment differences in the outcome of the two groups. Even if you know that treatment was random, you can't be sure about this common trend assumption. Actually, you rarely can be anyway but with pre-treatment data you can have at least an idea.

The issue I see with your approach is that you will not be able to see anything about the pre-treatment differences unless you have very precise information about the experiment or policy. It will be hard or even impossible to say something about the common trend assumption between the treatment and control groups which is a vital part of difference in differences.

For instance, say you have a job market program which is mandatory but in period 1 only motivated individuals will attend it. In period 2, which is the starting point of your data, the policy maker forces the other individuals to attend the job market program, and finally in period 3 you see all "treated" individuals. In this case it is hard to claim that those treated in period 1 and those in treated in period 2 have the same trend in their outcomes

1. due to the unobserved factors that led to treatment selection in the first round
2. due to the fact that individuals in period 1 have already been treated so their trend already changed (if the policy had an effect).

Of course this is a very artificial example and problematic mostly because treatment is non-random but I guess you will see the point. Without more knowledge about the experiment you can not credibly sell a difference in differences analysis in this set-up because you cannot say anything about the pre-treatment differences in the outcome of the two groups. Even if you know that treatment was random, you can't be sure about this common trend assumption. Actually, you rarely can be sure about it anyway but with pre-treatment data you can have at least an idea.