Dual orthogonal alignment of carbon nanotubes (CNTs) within the plane and perpendicular to a substrate is essential for many applications but difficult to obtain. Here, we demonstrate that it is possible using a combination of layer-by-layer deposition and ultrafast laser irradiation. Single-wall CNT-polymer composites preferentially aligned within the plane are irradiated with ultrafast laser pulses. After irradiation with distinct fluences at ambient conditions, morphology is seen where CNTs are formed into bundled CNTs with some orthogonal alignment. A model is presented to account for thermal expansion of the polymer and the formation of CNT bundles.