You'll need to check whether the spare outlet at the consumer unit is from the same MCB as the existing connector - if not, you'd need to fit another MCB.
Alternatively:
The other end of the existing outlet cable will connect to a Wago multi-block which you could use if it has a spare outlet.
Another option is to cut off the existing single plug and connect its cable, plus your new cable, to a new double-outlet Wago plug.
The Installation Datasheet is a useful guide if you haven't encounterd these connectors before.
I usually buy Wago connectors from here, and 1.5mm2 white flexible cable from Screwfix.