In previous work we have used geomagnetic activity as a proxy measure of the reconnection rate to determine how the upstream solar wind parameters control reconnection. Herein we extend that work to show how the solar wind magnetosonic Mach number, upstream solar wind beta, and the cone angle of the interplanetary magnetic field are correlated with a proxy measure of the efficiency of the reconnection process. Each of these three parameters controls the plasma beta in the magnetosheath downstream of the Earth's bow shock. The dependence of the efficiency of reconnection on the above parameters is in agreement with the expectation that large magnetosheath betas reduce the rate of or hinder the onset of dayside reconnection.