The unique emission properties of single-walled carbon nanotubes are attractive for achieving increased functionality in integrated photonics. In addition to being room-temperature telecom-band emitters that can be directly grown on silicon, they are ideal for coupling to nanoscale photonic structures. Here we report on high-efficiency coupling of individual air-suspended carbon nanotubes to silicon photonic crystal nanobeam cavities. Photoluminescence images of dielectric- and air-mode cavities reflect their distinctly different mode profiles and show that fields in the air are important for coupling. We find that the air-mode cavities couple more efficiently, and estimated spontaneous emission coupling factors reach a value as high as 0.85. Our results demonstrate advantages of ultralow mode-volumes in air-mode cavities for coupling to low-dimensional nanoscale emitters.