You can get $Tot(\mathcal{O}(n))$ in a similar manner by deleting a point from a weighted projective space. But this is more contrived. And is not really better than thinking of $Tot(\mathcal{O}(n))$ as a toric variety. So this probably is not what you want.

The tautological line bundle contains a divisor, isomorphic to $\mathbb{P}^n$, with normal bundle of degree $-1$ (namely, the zero section). On the other hand every effective compact divisor in $\mathbb{P}^{n+1}-\{x\}$ has a positive normal bundle. In particular, a hyperplane avoiding $x$ has normal bundle of degree $1$. The projection from the point identifies the complement of $x$ with the total space of this normal bundle.

Since the monodromy is always quasi-unipotent you can make a finite cyclic base change branched at zero, apply Clemens' description upstairs, and then analyze the effect of the cyclic group action.