Recent studies indicated that helical organic molecules, such as DNA and $alpha$-helical protein, can behave as Thouless quantum pumps when a rotating electric field is applied perpendicularly to their helical axes. Here we investigate the influence of long-range hoppings on this topological pumping of electrons in single-helical organic molecules. Under variation of the long-range hoppings governed by a decay exponent $mu$, we find an energy gap in the molecular band structure closes at a critical value $mu_c$ of the decay exponent and reopens for $mu$ deviating from $mu_c$. The relevant bulk bands in a pumping cycle acquire different Chern numbers in the strong ($mu<mu_c$) and weak ($mu>mu_c$) long-range hopping regimes, with a sudden jump at criticality. This topological phase transition is also shown to separate two distinct behaviors of the midgap end states in the pumping process. The end states carry quantized current pumped by the rotating electric field and the current forms a plateau by sweeping the Fermi energy over the gap. In the strong hopping phase, the quantized current plateau is positive, which is reversed to a negative one with smaller amplitude in the weak hopping phase. However, the reversal is a smooth crossover, not a sharp transition, due to the finite sizes of the molecules. We show that these transport characteristics of the topological phase transition could also be observed at finite temperatures.
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