If $\alpha < 1$, then $X$ has bounded variation and positive drift, and therefore it does not creep downwards; see Theorem 7.11 in Kyprianou's book Introductory Lectures on Fluctuations of Lévy Processes with Applications. This means that $$\mathbb{P}^x(X(\tau_{-1}) \ne -1) = 1,$$ for $x > -1$, where $$\tau_{-1} = \inf \{t > 0 : X(t) \leqslant -1\}.$$ In particular $$\mathbb{P}^x(X(\hat\zeta) \ne -1) = 1,$$ and consequently $$\mathbb{P}^x(\hat\zeta = \zeta) = 1.$$

If $\alpha < 1$, then $X$ has bounded variation and positive drift, and therefore it does not creep downwards; see Theorem 7.11 in Kyprianou's book Introductory Lectures on Fluctuations of Lévy Processes with Applications. This means that $$\mathbb{P}^x(X(\tau_{-1}) \ne -1) = 1,$$ for $x > -1$, where $$\tau_{-1} = \inf \{t > 0 : X(t) \leqslant -1\}.$$ In particular $$\mathbb{P}^x(X(\hat\zeta) \ne -1) = 1,$$ and consequently $$\mathbb{P}^x(\hat\zeta = \zeta) = 1.$$

Edited: As pointed out by kenneth, the above argument uses $\tau_{-1}$ defined with the weak inequality, while Theorem 7.11 in Kyprianou's book has a strict inequality, that is, $$\tau_{-1^-} = \inf \{t > 0 : X(t) < -1\}.$$ Getting from there to here is not immediate, though!

The proof of Theorem 7.11(i) involves the following reasoning: outside of an event of probability zero, $X(\tau_{-1^-}) = -1$ if and only if $\inf_{s \in [0, t]} X(s) = -1$ for some $t > 0$, and the latter event has zero probability due to the fact that the descending ladder-height process has no drift (see Theorem 5.9 in the book).

The same argument actually works for $X(\tau_{-1}) = -1$, and the proof is in fact slightly simpler. Unfortunately, I do not have time now to write up the details.