The original question suggests a certain lack of understanding by suggesting that using a thicker cable can somehow overload the 13A fuse. The cable itself has no bearing on the load applied to the fuse, only the equipment connected via that cable. In general, cost and physical issues aside, thicker cable is better than thinner cable, end of.
Buried behind furniture, under carpets
etc it will also run cooler, which is no bad thing either.
The safe current rating for any cable depends more on the insulation temperature rating than anything else- a silicone rubber insulated cable may well have a higher rating than a PVC insulated one for the same copper CSA. How and where it's routed also has a significant effect- look at the ratings for reeled vs unreeled extension leads.