Differentiable physics has recently been shown as a powerful tool for solving soft-body manipulation tasks. However, the differentiable physics solver often gets stuck when the initial contact points of the end effectors are sub-optimal or perform multi-stage tasks that are required to switch contact points, which often lead to many local minima. To address this challenge, we propose a contact point discovery approach (CPDeform) that guides the stand-alone differentiable physics solver to deform various soft-body Plasticine. The key idea of our approach is to integrate optimal transport-based contact points discovery into the differentiable physics solver to overcome the local minima from initial contact points or contact switching. On single-stage tasks, our method can automatically find the suitable initial contact points based on the transport priorities. On complex multi-stage tasks, we can iteratively switch the contact points of end-effectors based on transport priorities. To evaluate the effectiveness of our method, we introduce PlasticineLab-M that extends the existing differentiable physics benchmark PlasticineLab to eight new challenging multi-stage soft-body manipulation tasks. Extensive experimental results suggest that: 1) on multi-stage tasks that are infeasible for the vanilla differentiable physics solver, our approach discovers contact points that efficiently guide the solver to completion; 2) on tasks where the vanilla solver performs sub-optimally or near-optimally, our contact point discovery method performs better than or on par with the manipulation performance obtained with handcrafted contact points. Demos are available at our project page.