Recently Marcus, Spielman and Srivastava proved Weaver's KSr conjecture, which gives a positive solution to the Kadison-Singer problem. In [Coh16, Brä18], Cohen and Brändén independently extended this result to obtain the arbitrary-rank version of Weaver's KSr conjecture. In this paper, we present a new bound in Weaver's KSr conjecture for the arbitrary-rank case. To do that, we introduce the definition of (k, m)characteristic polynomials and employ it to improve the previous estimate on the largest root of the mixed characteristic polynomials. For the rank-one case, our bound agrees with the Bownik-Casazza-Marcus-Speegle's bound when r = 2 [BCMS19] and with the Ravichandran-Leake's bound when r > 2 [RL20]. For the higher-rank case, we sharpen the previous bounds from Cohen and from Brändén.