  We investigate eigenvector statistics of the Truncated Unitary ensemble
$\mathrm{TUE}(N,M)$ in the weakly non-unitary case $M=1$, that is when only one
row and column are removed. We provide an explicit description of generalized
overlaps as deterministic functions of the eigenvalues, as well as a method to
derive an exact formula for the expectation of diagonal overlaps (or squared
eigenvalue condition numbers), conditionally on one eigenvalue. This
complements recent results obtained in the opposite regime when $M \geq N$,
suggesting possible extensions to $\mathrm{TUE}(N,M)$ for all values of $M$.
