We offer a refinement of the classical Clifford inequality about special linear series on smooth irreducible complex curves. Namely, we prove about curves of genus g and odd gonality at least 5 that for any linear series $g^r_d$ with $d{leq}g+1$, the inequality $3r{leq}d$ holds, except in a few sporadic cases. Further, we show that the dimension of the set of curves in the moduli space for which there exists a linear series $g^r_d$ with d<3r for $d{leq}g+l,;0{leq}l{leq}frac{g}{2}-3$, is bounded by $2g-1+frac{1}{3}(g+2l+1)$.