Something much stronger is true:
Fix an arbitrary ordinal $\beta$ and a nondecreasing regressive map $f$ with domain $\beta-\{0\}$. Then there is a nondecreasing regressive map $d_f$ with the same domain such that $f\le d_f$ and $d_f$ takes on only finitely many values.
The proof is by induction on $\beta$. The base case $\beta=2$ is trivial, as is the successor case.
Now suppose the proof holds for all $\gamma<\lambda$ for $\lambda$ a limit, and take $\beta=\lambda$. Since $f$ is regressive and nondecreasing and $\lambda$ is a limit, there must be some $\gamma<\lambda$ such that $\forall \gamma<\eta<\lambda$, we have $f(\gamma)=f(\eta)$.
But now consider the map $F=f\upharpoonright \gamma+1. $ Since $\lambda$ is limit and $\gamma<\lambda$, $\gamma+1$ is also $<\lambda$, so we may apply the induction hypothesis to get a $d_{F}$. But now consider the map $d_f$ with domain $\lambda-\{0\}$ extending $d_{F}$ and taking on the value $d_{F}(\gamma)$ on all inputs $>\gamma$. This function has the desired properties, by assumption on $\gamma$.