This paper compares charge transport across self-assembled monolayers (SAMs) of n-alkanethiols containing odd and even numbers of methylenes. Ultraflat template-stripped silver (Ag-TS) surfaces support the SAMs, while top electrodes of eutectic gallium-indium (EGaIn) contact the SAMs to form metal/SAM//oxide/EGaIn junctions. The EGaIn spontaneously reacts with ambient oxygen to form a thin (similar to 1 nm) oxide layer. This oxide layer enables EGaIn to maintain a stable, conical shape (convenient for forming microcontacts to SAMs) while retaining the ability to deform and flow upon contacting a hard surface. Conical electrodes of EGaIn conform (at least partially) to SAMs and generate high yields of working junctions. Ga2O3/EGaIn top electrodes enable the collection of statistically significant numbers of data in convenient periods of time. The observed difference in charge transport between n-alkanethiols with odd and even numbers of methylenes - the "odd -even effect" - is statistically discernible using these junctions and demonstrates that this technique is sensitive to small differences in the structure and properties of the SAM. Alkanethiols with an even number of methylenes exhibit the expected exponential decrease in current density, J, with increasing chain length, as do alkanethiols with an odd number of methylenes. This trend disappears, however, when the two data sets are analyzed together: alkanethiols with an even number of methylenes typically show higher J than homologous alkanethiols with an odd number of methylenes. The precision of the present measurements and the statistical power of the present analysis are only sufficient to identify, with statistical confidence, the difference between an odd and even number of methylenes with respect to J, but not with respect to the tunneling decay constant, beta, or the pre-exponential factor, J(0). This paper includes a discussion of the possible origins of the odd-even effect but does not endorse a single explanation.