A signed graph $Σ=(G,σ)$ consists of an underlying graph $G=(V,E)$ with a sign function $σ:E\rightarrow\{-1,1\}$. Let $A(Σ)$ be the adjacency matrix of $Σ$ and $λ_1(Σ)$ denote the largest eigenvalue (index) of $Σ$.Define $(K_n,H^-)$ as a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we focus on the following problem: which spanning tree $T$ with a given number of pendant vertices makes the $λ_1(A(Σ))$ of the unbalanced $(K_n,T^-)$ as large as possible? To answer the problem, we characterize the extremal signed graph with maximum $λ_1(A(Σ))$ among graphs of type $(K_n,T^-)$.