While lineage claims that the update process is mostly automatic on the Galaxy S III (gt-9300i), I found that after downloading an update and rebooting I ended up in my TWRP boot screen, with nothing much automatic going on at all.
It took me a while to determine the storage location of the downloaded updates (that much worked I’m happy to report), and it seems that for me (LineageOS 14.1) they end up on the internal SD card in a directory called
The trick was to use the “up one level” function in TWRP’s image selection to actually see the root level of the internal SD card.
Then, all I needed to do was to select the most recent update (it’s in zip format so you may need to switch the image / zip display in TWRP’s “install” screen, flash it (apparently you don’t have to wipe the cache for it to work), reboot the system and wait about 15 minutes, and hey presto!, here’s my most recent Lineage OS update running fine on my S III.
Caveat: During the initial migration from CM13 to Lineage 14.1, I found the system would not boot properly if I chose to select the “SuperSU” package as offered by the Lineage OS installer. Not choosing to install that package would allow the system to boot properly.
I hope this helps you updating and keeping current your beloved GT-9300i phone!