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Context . The outflow from the Class 0 protostar IRAS 04166+2706 (hereafter IRAS 04166) contains a remarkably symmetric jet-like component of extremely high-velocity (EHV) gas. Aims . We studied the IRAS 04166 outflow and investigated the relation between its EHV component and the slower outflow gas. Methods . We mosaicked the CO(2–1) emission from the IRAS 04166 outflow using the 12m and the Compact Arrays of ALMA. We also developed a ballistic toy model of the gas ejected laterally from a jet to interpret the data. Results . In agreement with previous observations, the ALMA data show that the slow outflow component is distributed in two opposed conical lobes and has a shear-flow pattern with velocity increasing toward the axis. The EHV gas consists of a series of arc-like condensations that span the full width of the conical lobes and merge with their walls, suggesting that the fast and slow outflow components are physically connected. In addition, position–velocity diagrams along the outflow axis show finger-like extensions that connect the EHV emission with the origin of the diagram, as if part of the EHV gas had been decelerated by its interaction with the low-velocity outflow. A ballistic model can reproduce these finger-like extensions assuming that the EHV gas consists of jet material that has been ejected laterally over a short period of time and has transferred part of its momentum to the surrounding shear flow. Conclusions . The EHV gas in the IRAS 04166 outflow seems to play a role in the acceleration of the slower gas component. The presence of similar finger-like extensions in the position–velocity diagrams of other outflows suggests that this process may be occurring in other systems, even if the EHV component is not seen because it has an atomic composition.