We report the superconducting properties of new hexagonal Nb$_{10+2x}$Mo$_{35-x}$Ru$_{35-x}$Rh$_{10}$Pd$_{10}$ high-entropy alloys (HEAs) (0 $leq$ $x$ $leq$ 5). With increasing $x$, the superconducting transition temperature $T_{rm c}$ shows a maximum of 6.19 K at $x$ = 2.5, while the zero-temperature upper critical field $B_{rm c2}$(0) increases monotonically, reaching 8.3 T at $x$ = 5. For all $x$ values, the specific heat jump deviates from the Bardeen-Cooper-Schreiffer behavior. In addition, we show that $T_{rm c}$ of these HEAs is not determined mainly by the density of states at the Fermi level and would be enhanced by lowering the valence electron concentration.
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