That was never my impression. At the very beginning of the movie we don't know exactly where the Enterprise is, other that they are "three hours twenty-five minutes" from the Typhon sector, wherever that is. The Typhon sector is where Admiral Hayes is mobilizing a fleet. The Enterprise then travels to the RNZ and has time to complete a sensor sweep of the RNZ. At this point the Fleet (presumable in the Typhon sector) engages the Borg. The Enterprise at that point sets course for Earth at maximum warp, the Borg cube is still engaged with the fleet. The cube and the fleet arrive at Earth only shortly before the Enterprise does. The Enterprise might have spent multiple days at maximum warp to reach Earth, in order to intercept the Borg cube. From the RNZ, the Enterprise might have been incapable of joining the fleet at any other point other than Earth.