In which type of Hamiltonian does the ansatz need an entangler in VQE?

I would like to prepare an ansatz for VQE. I know UCC ansatz is one of the most useful ansatz, but I would like to reduce the number of CNOT.

From the matrix of a given Hamiltonian, can we know the ansatz for the given Hamiltonian needs entanglers or not?

I think that $$UCC$$ ansatz is good for chemistry related problems. However, if you an arbitrary Hamiltonian and want to find its smallest eigenvalue then it's not very obvious what the anstaz form should look like.... it will also depend on how you initialize your state. And if you define an anstaz with polynomial number of gates, you don't expect to be able to explore the entire Hilbert space.
$$H = \sum h_i \sigma_i + \sum h_{ij}\sigma_{i}\sigma_{j} + \cdots$$