Can the thrust of a turbofan be increased by increasing bypass ratio?
GE used the core of the F101 engine as the core of the CFM56 (with some modifications, presumably). The core of a turbofan consists of the high pressure compressor, the combustion chamber and the high pressure turbine. In this case, we talk about a 9-stage HPC and a single-stage HPT. Everything else was newly developed for the high-bypass turbofan.

Unlike just putting a larger fan on the low-pressure shaft, this is possible becasue the high-pressure components and the low-pressure components are relatively independent in a turbofan, i.e. the core of the engine is not affected too much by the size of the fan, the bypass ratio or other changes in the low-pressure area.

The new low pressure turbine (4-stage instead of 2-stage) can operate at a lower rpm and transmit more of the exhaust gas energy from the engine core to the larger fan (the exhaust gas coming out of the new LPT is probably cooler and slower than that coming out of the F101 LPT). The new low-pressure compressor increases the overall pressure ratio and therefore increases the amount of energy that goes to the low pressure compressor and fan, thereby improving the overall efficiency. This is probably possible because the non-supersonic operational envelope of the CFM56 means that intake air total temperatures will be much lower so that the same EGT margin can be achieved with a higher pressure ratio. In other words, becasue the incoming air is cooler, it can be compressed further without damaging the high-pressure turbine because it is too hot.

The static thrust of the CFM56 is 19500 lbf, which is up about 15 % from the F101's 17000 lbf without the afterburner. While a small part of this is probably an aerodynamic loss due to the unused afterburner chamber, most of it has to come from the better use of the energy offered by the engine core exhaust in the larger fan. The difference might even be higher becasue the maximum thrust of the F101 might be adjusted for a much lower mean time between shop visits. I'm not sure about that though.

This shows that the main benefit of the high-bypass low pressure section of the engine is a significantly increased fuel efficiency, an increase in the static thrust and (much) lower noise levels. The F101, on the other hand, can operate at higher speeds, produce more thrust at high speeds, is lighter and has a much smaller cross-section.

(Edited per comment by fooot, thanks for finding the source.)