I am aware of this:
However, this only gives you the borders for, for example, 'L' rather than:
L1, L10, L11, L12 etc.
Can I obtain these fine-grained border data from somewhere free?
Take a look at this site: UK Postcodes
It has the CSV data for all the boundaries. You can also download a KML file at each level to display in Google Earth, or QGIS where you can do more analysis.
Unfortunately you may need to individually download each Area to get detail for the next level down. For example, select CH from the list and then click 'Download the individual postcode data in KMl format' to get data of the next level down (CH1, CH2 etc).
I found KML files for district level boundaries for all UK Postcode areas on github at https://github.com/missinglink/uk-postcode-polygons/tree/master/kml
have tested with one file on google maps and it overlays the boundaries correctly