Capillary rise exceeding tube height

No, the water does not overflow. If the tube is shorter than the calculated rise, the meniscus simply becomes less curved, so that the pull of surface tension is still balanced by the weight of the (shorter) column.

For a tube of radius $r$ with contact angle $\theta$, equating the vertical surface-tension pull to the weight of the column gives

$$2\pi r\sigma\cos\theta=\rho g\,\pi r^{2}h\quad\Rightarrow\quad h=\frac{2\sigma\cos\theta}{\rho g r}$$

If the tube height is $h_t<h$, the water rises to the top and the meniscus at the rim adjusts to a new apparent angle $\theta'$ (larger than $\theta$) such that

$$\cos\theta'=\frac{\rho g r\,h_t}{2\sigma}<\cos\theta$$

Equivalently, the radius of curvature of the meniscus becomes $R'=r/\cos\theta'$, which is larger than before (a flatter meniscus), so the vertical pull $2\pi r\sigma\cos\theta'$ just equals the weight of the shorter column. The column height stays $h_t$ and the water stays in the tube. However short the tube, the meniscus only flattens further ($\theta'\to90^\circ$ as $h_t\to0$); surface tension at the rim always holds the water, so it never flows over.