In this paper, the following problem is considered: does there exist a $t$-arc-transitive regular covering graph of an $s$-arc-transitive graph for positive integers t greater than $s$? In order to answer this question, we classify all arc-transitive cyclic regular covers of the dodecahedron graph. Two infinite families of 3-arc-transitive abelian covering graphs are given, which give more specific examples that for an $s$-arc-transitive graph there exist $(s+1)$-arc-transitive covering graphs.