In this paper, we present some new improvements of dynamic Opial-type inequalities of first and higher order on time scales. We employ the new inequalities to prove several results related to the spacing between consecutive zeros of a solution and/or a zero of its derivative of a second-order dynamic equation with a damping term. The main results are proved by making use of a recently introduced new technique for Opial dynamic inequalities, the time scales integration by parts formula, the time scales chain rule, the time scales Taylor formula, and classical as well as time scales versions of Hölder's inequality.