The best known upper bound for the critical value $\lambda_c$ of the basic one dimensional contact process is 2. Most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds. This seemed not to be the case for the approach which yielded $\lambda_c \leq 2$ for the basic contact process. In this paper, we propose a technique for generating better bounds in this context. To illustrate its use, we carry out the full program in one case, with the conclusion that $\lambda_c \leq 1.942$.