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Your title asks how to set the last state of a trajectory as the start state of a new plan, which you probably already figured out (no guarantee on the syntax):

moveit::planning_interface::MoveGroupInterface::Plan my_plan;
move_group_interface.plan(my_plan)
robot_state::RobotState last_joint_state = my_plan.trajectory_.getLastWayPoint();
move_group_interface.

But in your question you ask how to set the start state of a robot motion. Your approach looks fine at first sight. I would check if setFromIK is failing for start_state_lift, and if the joint state you receive from that call is actually the same as the state of your robot after you executed the motion. You can use moveit_visual_tools to visualize the state.