A tire having a carcass reinforcement surmounted radially by a crown reinforcement having at least two working plies, the width Lt of which is equal to the width of the axially narrowest working ply, and surmounted radially by a tread (or “cap”) and interposed radially between the cap and the crown reinforcement by a sublayer (or “base”), this base extending axially and radially under all of the cap and radially on the outside of the crown reinforcement over a total width at least equal to the width Lt of the crown reinforcement, an equatorial plane dividing the cap axially into an outer half cap (TE) and an inner half cap (TI) of the same axial width, the outer half cap (TE) positioned axially towards the outside of a vehicle and the inner half cap (TI) positioned axially towards the inside of the vehicle, the base comprising an additional volume radially under the inner half cap (TI), such that the volume occupied by the base radially under the inner half cap (TI) is greater than the volume occupied by the base radially under the outer half cap (TE), and the material of the base is a rubber-based composition having hysteresis lower than the hysteresis of the rubber-based material of the cap.