Robots often have to perform manipulation tasks in close proximity to people. As such, it is desirable to use a robot arm that has limited joint torques to not injure the nearby person and interacts with the environment to explore new possibilities for completing a task. By bracing against the environment, robots can expand their reachable workspace, which would otherwise be inaccessible due to exceeding actuator torque limits, and accomplish tasks beyond their design specifications. However, motion planning for complex contact-rich tasks requires reasoning through the permutations of different possible contact modes and bracing locations, which grow exponentially with the number of contact points and links in the robot. To address this combinatorial problem, we developed INSAT, which interleaves graph search to explore the manipulator joint configuration and the contact mode space with incremental trajectory optimizations seeded by neighborhood solutions to find a dynamically feasible trajectory through contact. In this paper, we present recent additions to the INSAT algorithm that improve its runtime performance. In particular, we propose Lazy INSAT with reduced optimization rejection that systematically procrastinates its calls to trajectory optimization while reusing feasible solutions that violate boundary constraints. The algorithm is evaluated on a heavy payload transportation task in simulation and on physical hardware. In simulation, we show that Lazy INSAT can discover solutions for tasks that cannot be accomplished within its design limits and without interacting with the environment. In comparison to executing the same trajectory without environment support, we demonstrate that the utilization of bracing contacts reduces the overall torque required to execute the trajectory.