White Giant only features in the second Red Dwarf novel, Better Than Life, and is never mentioned in the television series. It's never explained what the craft looks like. It is probably named after White Midget.
White Giant appears more prominently in the second novel, Better Than Life, although it is little described beyond it's colour and large size. Arnold Rimmer and Cat use White Giant to find Lister on Garbage World. Given that Starbug was destroyed by acid rain, and the destruction of Blue Midget, White Giant was left as the only remaining shuttlecraft for Red Dwarf. However, in the two sequel novels Starbug has been rebuilt and White Giant never appears.
White Giant appears in early scripts, but do not actually appear in the television series. One explanation of the absence of the original White Giant is that the Cat species likely modified them into the Cat Arks, and left Red Dwarf with them, a decade or so before Lister came out of stasis. ("Waiting for God", Series I)
It is likely that White Giant makes an appearance in Series VIII as the unnamed transport shuttle used by 'The Canaries', the so-called prisoner suicide squad who attempt dangerous missions from The Tank on Red Dwarf. Given that it can transport numerous squads at once, this craft was almost certainly larger than Starbug, and of course was white. This may have been intended to be a nanobot-rebuilt and upgraded White Giant, as with Series VIII's Red Dwarf itself, and also with Series VIII's Blue Midget.
The Canary shuttle was not seen to be used by the crew to evacuate and abandon Red Dwarf at the end of "Only the Good...". However, since the shuttle has not been seen again, it likely that it too evacuated. The Canaries are referenced in the Series X episode "Fathers & Suns", when Lister is in an airlock and a Canary uniform can be seen among the spacesuits.