This paper presents a novel test point insertion (TPI) method for a pseudo-random built-in self-test (BIST) to reduce the area overhead. Recently, a new TPI method for BISTs was proposed that tries to use functional flip-flops to drive control test points instead of adding extra dedicated flip-flops for driving control points. The replacement rule used in a previous work has limitations preventing some dedicated flip-flops from being replaced by functional flip-flops. This paper proposes a logic cone analysis-based TPI approach to overcome the limitations. Logic cone analysis is performed to find candidate functional flop-flops for replacing dedicated flip-flops. Experimental results indicate that the proposed method reduces the test point area overhead significantly with minimal loss of testability by replacing the dedicated flip-flops.