Further Strengthening Supply Chains for Servicemembers and Security Act of 2023 This bill requires the Defense Logistics Agency to modify Defense Logistics Agency Instructions 5025.03 and 3110.01 to require Defense Logistics Agency Troop Support to coordinate annually with customers in the military departments to test responsiveness of the agency's contingency contracts for pharmaceuticals and to include the results of the testing in the annual Warstopper Program reports.