probably a bit late jusging by the completion date, but in case anyone else has this question I will answer anyway.
If your deal, which you state is variable, is a _base_ rate tracker product, then your rate will automatically go down with the _base_ rate. In this case there is no reason to go else where. In all probability lots of lenders will only pass part of this cut on to new customers by slightly increasing their margins on tracker rates. (and this certainly seems to be the case at the moment)
If your deal is just on the lenders SVR (and going on the information you have given I don't think this is the case) then there is no guarantee that the lender wil reduce their SVR and you may wish to take a quick look around. However, as you are one day away from completion if you change lender now you may well lose some or all of any fees you have paid.
As always, the best advice is to speak to a good independent
mortgage broker.