Suppose that we are given two vertex covers \(C_{0}\) and \(C_{t}\) of a graph G, together with an integer threshold \(k\ge \max \{\left| C_0 \right| , \left| C_t \right| \}\). Then, the vertex cover reconfiguration problem is to determine whether there exists a sequence of vertex covers of G which transforms \(C_{0}\) into \(C_{t}\) such that each vertex cover in the sequence is of cardinality at most \(k\) and is obtained from the previous one by either adding or deleting exactly one vertex. This problem is PSPACE-complete even for planar graphs. In this paper, we first give a linear-time algorithm to solve the problem for even-hole-free graphs, which include several well-known graphs, such as trees, interval graphs and chordal graphs. We then give an upper bound on \(k\) for which any pair of vertex covers in a graph G has a desired sequence. Our upper bound is best possible in some sense.