The Ubuntu GIS team has GDAL 2.2.1 for Xenial and has PostGIS 2.3.3 compiled against it. From the PostGIS build logs
checking for gdal-config... /usr/bin/gdal-config
checking GDAL version... 2.1.3
Install with
sudo add-apt-repository ppa:ubuntugis/ppa;
sudo apt-get update;
sudo apt-get install postgis;
EDIT by OP:
I have tried ubuntugis before and it didn't work, because I used the newest PostgreSQL version (9.6) from the Postgresql repository, but as it turns out, the current ubuntugis-unstable PostGIS version is compiled against PostgreSQL 9.5. So I removed the Postgresql repository from my sources.list
and did:
sudo add-apt-repository ppa:ubuntugis/ubuntugis-unstable
sudo apt-get update
sudo apt-get install postgresql-9.5
sudo apt-get install postgresql-9.5-postgis-2.3 postgresql-contrib-9.5 postgresql-9.5-postgis-scripts
sudo apt-get install postgis
That did the trick for me, just to give a little more context.