Modifying a TLC mobile phase in order to increase the RF values
Question: Would the hexane and ethyl acetate be considered the mobile phase/solvent?

Answer is yes: Hexane/ethyl acetate systems ($0$ to $100\%$ ethyl acetate in hexane) are frequently used mobile phases for TLC technique. However, as Waylander pointed out in his comment:

The type of mobile phase really depends on what the substrate and product are. It is worth running half a dozen trial mobile phases to see which one gives a decent separation.

Also, it is  true, as Buttonwood pointed out, a high $R_f$ value is not very useful if $\Delta R_f$ between the two compounds (or more in the mixture) of interest is small unless it increase upon change of mobile phase. For example, TLC of the mixture of benzoic acid and benzocaine (ethyl 4-aminobenzoate) shows no separation in $50\%$ ethyl acetate in hexanes. However, when mobile phase is change to $5\%$ methanol in dichloromethane, it has shown $\gt 33\%$ separation in $R_f$. So it is a trial and error method, but worth trying.