Ising machines, hardware accelerators for combinatorial optimization and probabilistic sampling problems, have gained significant interest recently. A key element is stochasticity, which enables a wide exploration of configurations, thereby helping avoid local minima. Here, we refine the previously proposed concept of coupled chaotic bits (c-bits) that operate without explicit stochasticity. We show that augmenting chaotic bits with stochasticity enhances performance in combinatorial optimization, achieving algorithmic scaling comparable to probabilistic bits (p-bits). We first demonstrate that c-bits follow the quantum Boltzmann law in a 1D transverse field Ising model. We then show that c-bits exhibit critical dynamics similar to stochastic p-bits in 2D Ising and 3D spin glass models, with promising potential to solve challenging optimization problems. Finally, we propose a noise-augmented version of coupled c-bits via the adaptive parallel tempering algorithm (APT). Our noise-augmented c-bit algorithm outperforms fully deterministic c-bits running versions of the simulated annealing algorithm. Other analog Ising machines with coupled oscillators could draw inspiration from the proposed algorithm. Running replicas at constant temperature eliminates the need for global modulation of coupling strengths. Mixing stochasticity with deterministic c-bits creates a powerful hybrid computing scheme that can bring benefits in scaled, asynchronous, and massively parallel hardware implementations.