Demolition is sadly fully underway now. I believe they started last week and it's estimated to take 40 weeks to clear the site. I'd highly recommend visiting the building before it's destroyed, there's a lot to see there aren't a whole lot of similar places left to close. Access is going to be easy and you can just casually walk through the dock gate into the site.
These photos are taken from the 'Southampton Heritage Photos' Facebook group:
Would be nice if they pulled off all of the crappy cladding from Pre-war silo before tearing it down, the brickwork looks great in old photos
May as well show an update now it's in public, again from the Southampton heritage group. Looks like only the front block remains, the belt manlift is clearly visible on the left of the skinned section.
Just to let you know what you have called multi purpose warehouse. It was know as the HOVIS warehouse when we cooked germ that went into Hovis bread and a small stone ground mill.