A question about paired sample t test?

Your help on the following question is appreciated. Thanks.

I know paired-sample t test is for the same group of people to test their pre and post tests and see if they have a significant gain between their pre and post tests. If the treatment is progressive, can I use the same people to collect data over time and run paired-sample t test together because the sample size is very small (only 6 persons)?

Year 1: Treatment 1 with 3 persons' (ABC) pre and post tests
Year 2: Treatment 2 (improvement of treatment 1) with the same 3 person's (ABC) pre and post tests
Year 3: Treatment 2 (improvement of treatment 1) with the other 3 persons' (DEF) pre ad post tests

In this case, can I run paired-sample t test for 9 data points (3 data points [ABC] from Year 1, 3 data points [ABC] from Year 2, and 3 data points [DEF] from Year 3) and then report the effect of treatment 1 in general (since treatment 2 is based on the improvement of treatment 1)?

I know some people might argue that I use the same people twice (ABC) in year 1 and year 2. But can I argue that paired-sample t test is concerned with the "gain" between pre and post tests. Thus, although they might have a higher score for the pre test in the second year, we expect them to have a higher score for the post test too. That is, we expect ABC to have a "gain" in both Year 1 and Year 2.

Your insights and advices are much appreciated. References are much appreciated too.