Let G be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of G is such a path in which no two edges have the same color. Let the color degree of a vertex v to be the number of different colors that are used on edges incident to v, and denote it by \(d^c(v)\). In a previous paper, we showed that if \(d^c(v)\ge k\) (color degree condition) for every vertex v of G, then G has a rainbow path of length at least \(\lceil (k+1)/2\rceil \). Later, in another paper we first showed that if \(k\le 7\), G has a rainbow path of length at least \(k-1\), and then, based on this we used induction on k and showed that if \(k\ge 8\), then G has a rainbow path of length at least \(\lceil (3k)/5\rceil +1\). In 2010, Gyarfas and Mhalla showed that in any proper edge-colored complete graph \(K_n\), there is a rainbow path with no less than \((2n+1)/3\) vertices. In the present paper, by using a simpler approach we further improve the result by showing that if \(k\ge 8\), G has a rainbow path of length at least \(\lceil (2k)/3\rceil +1\).