It seems my old GPS device can achieve greatly improved precision by simply switching on the "WAAS mode" in settings - even in Europe.
OK, we know that GPS works by receiving time stamps from different satellites and by comparing the delay in the signals from the different satellites it can calculate its distance to the different satellites and therefore its own position here on the ground.
But different atmospheric phenomenons can cause the radio signal to travel slightly faster or slower than usual, causing degraded accuracy. To compensate for this, there are observation stations around the world that analyzes the current error in the GPS system in that part of the world and then sends this information back to the satellites - the satellites then sends this "augmentation" information to all GPS receivers the same way that the normal GPS signal is sent. A compatible GPS device checks if there is augmentation information available for the area where the GPS device is, and if so, includes the augmentation data in its calculations. This is also known as "Differential GPS" or "Satellite-Based Augmentation Systems" (SBAS).
This brings us to the "compatible device" part.
In the US, this system has been in use since 2003 and is called "WAAS", or "Wide Area Augmentation System". Many ordinary GPS devices for consumers are WAAS compatible, including my Garmin Foretrex 101 which I bought in 2004. Usually, WAAS support is disabled and needs to be specifically switched on. My Garmin was in "Normal" mode and one of the alternatives was "WAAS mode". Most texts about WAAS states that it is available only in the US which is correct but misleading. Some texts also state that Europe has a "similar" system under development, known as "EGNOS" or "European Geostationary Navigation Overlay System".
As it turns out, EGNOS and WAAS are fully compatible and EGNOS is operational. So I switched to WAAS mode and sure enough, after a few minutes the ordinary "Accuracy" changed to "DIFF" to show that the device is in "differential GPS" mode and making use of augmentation data. Based on the satellite list, it seems to have received augmentation data from EGNOS satellite AOR-E (prn 120) listed as satellite 33.