Difference-in-differences (DiD) with policy change during post-period

In your setting, it is natural to extend the difference-in-differences (DiD) setup. There’s no problem with having more than one “treatment $\times$ post” interaction term, especially if the second reform happens later. In fact, it’s often preferable to keep the sample intact rather than drop years and lose power, as you already noted.

Concretely, your equation would look like the following:

$$ Y_{ist} = \gamma_s + \lambda_t + \delta_1 (C_{s} \times Post_{t = 2018})+ \delta_2 (C_{s} \times Post_{t=2020}) + \epsilon_{ist}, $$

where $C_{s}$ is the indicator for being in California (vs. your chosen control group). The variable $Post_{t = 2018}$ equals 1 in the years after first policy change, 0 before. Likewise, the variable $Post_{t=2020}$ equals 1 in years after the second policy change (policy modification), 0 before. As is customary in most DiD applications, $\gamma_s$ and $\lambda_t$ denote the unit and time fixed effects, respectively. Here, $\delta_1$ is the effect of the first reform relative to the pre-2018 baseline, and $\delta_2$ captures any incremental effect of the second reform relative to the 2018 regime. Put simply, your model will allow you to estimate the effect of each policy change separately, while still using all available data.

Regarding year fixed effects, you definitely should include them. Year fixed effects absorb any time-varying shocks common to all groups (e.g., national labor market shifts, COVID lockdowns). I would also think carefully about the choice of control groups. If you use “mothers of older children in California” as the control, you are leveraging within-state untreated groups. This naturally lends itself to a difference-in-difference-in-differences (DDD) design, since one group (mothers of infants) is directly exposed to the PFL reform while another group in the same state (mothers of older children) is not. If instead you use mothers of infants in other states that are completely unaffected by California’s PFL program, then you fall back on a more standard DiD across states, relying on the California $\times$ post interactions in the earlier equation.

The short answer is yes. Adding separate interactions for each reform year is not only acceptable but often the recommended way to structure a DiD when you have sequential reforms.