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I don't see the need to rip up more floor. The new joist will be supported at both ends. One at the existing joist / sleeper wall, and the other on a new cross joist / wall plate at the front of the chimney breast. It's just quite how and where we attach the latter that's undecided.
In other news, I got a free belt sander today, just because the previous owner didn't think to change the belt and though it was broken. Win.
Surely the thing to do is rip up the whole floor and buy a joist that's 1' longer?
Nah, bolting an extra section to the current joist should do it, but you'll probably want it to be 3x as long as the unsupported section so you can get it nice and secure, otherwise squeaky floorboards 4lyf no?