The 2-site TDVP algorithm modifies MPS bond-dimensions on the fly, speeding up when entanglement is low. However, this comes at the cost of slowing down simulations when entanglement is high. The most time consuming part of the TDVP is solving the Schrödinger Equation for its effective Hamiltonians. In the 2-site variant, the effect of this is made worse by the increased dimension of combined 2-site tensors. However, limiting the time evolution to 1-site tensors complicates deciding when to grow the bond dimensions. This article presents a new version of the TDVP algorithm, inspired by the existing A1TDVP algorithm which enables dynamic bond dimensions while avoiding to solve 2-site SEs. Like the A1TDVP, the new algorithm is also capable of changing bond dimensions on the fly, without solving SEs on combined tensors. By relying on Singular Value Decompositions and approximate time evolutions, accurate estimations for good bond dimensions can be calculated.