We propose a method to modulate the bandgaps in narrow-gap carbon nanotubes using a transverse electric field. Unlike previous investigations, we include curvature effects of the nanotubes by incorporating both π- and σ-orbitals in our tight-binding calculations. The calculations show that the narrow curvature-induced bandgaps decrease quadratically with electric field amplitude to zero. As the electric field amplitude continues to increase, the bandgap then expands in a similar manner to that presented in earlier studies on metallic nanotubes. The bandgap dependence is verified by analytical calculations, which also agree with preceding analyses for the limit of no curvature.