In order to achieve warp speed, one does not need to have a specialized rocket. Instead, one needs to find a substance that carries a negative density energy. This is something that can be predicted by theory of general relativity. In the universe, there do exist elements that suggest such a property.
For instance, empty space caries a negative density energy. This is something that is being investigated by NASA currently. Hence, achieving warp speed is not dependent on the speed of the rocket itself. Rather, it’s a complex topic. The big question whether humans will ever be able to achieve warp speed still remains unanswered.