EDIT: The most elegant solution can be found here:
I found a solution as a combination of this answer and changing the PYTHONPATH in two of my config-files:
/home/user/.bashrc
/home/user/.bash_profile
Nomenclature: The initial (lower) python-version will be represented by 3.x, the newer one by 3.y.
NOTE on the necessity of previously modifying the PYTHONPATH:
In my case, as there were many python-versions installed in various ways and locations, it was needed to prepend the target sitepackage directory to every other one present in the entire PYTHONPATH, since only this way the extracted python-modules will be installed in the correct directory via python -m pip install --user -r packages_python_3_x.txt.
If on your system, the PYTHONPATH points directly to the site-packages of desire, i.e. they are in the first position of your entire PYTHONPATH, you wouldn't need to modify it.
Moreover, if you're working with virtual-environments (e.g. pyenv) or other environment managers, you will be able to separate each site-packages associated to every virtual-environment automatically. Then, there should also be no need to modify the PYTHONPATH. By contrast, I didn't work with virtual, local and global python environments and that's why it was necessary for me to prepend the site-package-directory of interest to the PYTHONPATH. Now, I'm working with virtual-environments for different projects, which I recommend to everyone.
1.) Preparing the PYTHONPATH (if needed)
Continuing with said config-files, therein, insert the following line:
PYTHONPATH="/home/linuxbrew/.linuxbrew/lib/python3.x/site-packages:$PYTHONPATH"
In case there was a similar line pointing to the previous python-version directory, replace that with the most recent one in accordance with your situation (e.g. 3.7 with 3.8 in the directory path).
Moreover, if the system python3-path is prepended to the PYTHONPATH, comment that comment out in order to give your python distribution (be it homebrew, anaconda, or whatever you like) precedence.
# NOTE on the following commando: it is commented out on purpose (you could also delete it from the config-files, if you don't consider to use these site-packages again, but I kept it for playing safe)
#export PYTHONPATH="/usr/lib/python3/dist-packages:$PYTHONPATH"
In order to make changes take effect immediately, carry out the following commandos in your UNIX-terminal:
source /home/user/.bash_profile
source /home/user/.bashrc
2.) Extract currently installed modules and reinstall them with new version 3.y
Finally, extract all python-packages installed to a textfile. In my case, this had been done with pip, so this would be:
python -m pip freeze --path /home/linuxbrew/.linuxbrew/lib/python3.x/site-packages > packages_python_3_x.txt
The textfile should contain lines like so:
absl-py==0.11.0
access==1.1.1
affine==2.3.0
aiohttp==3.7.2
aiohttp-cors==0.7.0
...
Finally, (re)install all these packages into the latest site-packages folder (here done using pip):
python -m pip install --user -r packages_python_3_x.txt
The package manager pip should choose the correct folder since you have updated the PYTHONPATH previously.
The new site-packages directory in this example should be /home/linuxbrew/.linuxbrew/lib/python3.y/site-packages.