Since the default file manager does not seem to support what I want, I looked into alternative file managers and installed Nemo (note that this is more difficult at the moment, see below). It is similar enough while having many more features and not displaying the above mentioned annoying issue (at least for version 1.x, see below).
Alternatively, PCManFM seems to actually not have this issue. The name column won't get truncated by default, while allowing manual resizing. It does not have an as beautiful the UI though (but still good enough).
Installing Nemo on Ubuntu 13.10 is not as easy. There exists a PPA which provides a seemingly compatible version of Nemo 2.0 for Saucy Salamander (ppa:gwendal-lebihan-dev/cinnamon-stable). It worked fine at first, being more useful than the default Nautilus, although still having this issue but only with even narrower windows. However, this will also install Cinnamon 2.0. And this currently breaks Unity in 13.10. I experienced that after a restart, where I was not able to login before removing all cinnamon/nemo packages first. So beware!
The otherwise official repo at https://launchpad.net/~noobslab/+archive/nemo provides Nemo (version 1.x, currently 1.7.4) for Ubuntu. As of now there is no version for saucy, however. But I was able to install the version for raring by downloading the deb packages for:
All but the actual nemo package installs fine. Nemo has a dependency on libgnome-desktop-3-4 but Ubuntu 13.10 has libgnome-desktop-3-7. So what I did was extract the deb package, modify the DEBIAN/control file, repackage it and install it. As a last step I had to link the dynamic library for 3-7 to 3-4 because Nemo expects this library.
# extract the raring package
dpkg-deb -x ../nemo.deb .
dpkg-deb -e ../nemo.deb
# modify libgnome-desktop dependency from 3-4 to 3-7
dpkg-deb -b . ../nemo~saucy.deb
# link the libraries
sudo ln -s /usr/lib/libgnome-desktop-3.so.7 /usr/lib/libgnome-desktop-3.so.4
# install nemo~saucy.deb