Conventional neoclassical transport theory is extended to include the effects of orbit squeezing, and to allow the effective poloidal Mach number UpM=[(V∥/vt)+(VEB/vtBp)] of the order of unity for incompressible tokamak plasmas. Here, V∥ is the parallel mass flow, vt is the ion thermal speed, VE is the poloidal E×B drift speed, B is the magnetic field strength, and Bp is the poloidal magnetic field strength. It is found that ion thermal conductivity is reduced from its conventional neoclassical value in both banana and plateau regimes if UpM>1 and S>1. Here, S=[1+cI2Φ′′/(Ω0B0)] is the orbit squeezing factor with c the speed of light, I=RBt, R the major radius, Φ the electrostatic potential, B0 the magnetic field strength on the axis, Ω0=eB0/Mc, M the ion mass, e the ion charge, Φ′′=d2Φ/dψ2, and ψ the poloidal flux function. However, there is an irreducible minimum for the ion thermal conductivity in the banana-plateau regime set by the conventional Pfirsch–Schlüter transport.