Suppose α is an orientation-preserving diffeomorphism (shift) of $${\mathbb {R}_+=(0,\infty)}$$ onto itself with the only fixed points 0 and ∞. In Karlovich et al. (Integr Equ Oper Theory 2011, doi: 10.1007/s00020-010-1861-0 ) we found sufficient conditions for the Fredholmness of the singular integral operator with shift $$(aI-bW_\alpha)P_++(cI-dW_\alpha)P_-$$ acting on $${L^p(\mathbb {R}_+)}$$ with 1 < p < ∞, where $${P_\pm=(I\pm S)/2, S}$$ is the Cauchy singular integral operator, and $${W_\alpha f=f\circ\alpha}$$ is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α′ of the shift are bounded and continuous on $${\mathbb {R}_+}$$ and may admit discontinuities of slowly oscillating type at 0 and ∞. Now we prove that those conditions are also necessary.