Can I add insulation over the original layer in my new home's basement?

Really depends on your location. Aside from removing the wrap insulation and doing a spray foam insulation (which is flammable), the best job includes tar paper on the concrete(so any moisture coming through the wall doesn't wet the insulation) then the insulation in the walls with the bottom 6 inches left out and a 6 mm vapour barrier stopping short of the floor. The reason why the gap is so air can get into the wall and the moisture can get out. Also be sure to put in adequate vents (including cold air returns) to ensure good air flow.