  Hadamard ideals were introduced in 2006 as a set of nonlinear polynomial
equations whose zeros are uniquely related to Hadamard matrices with one or two
circulant cores of a given order. Based on this idea, the cocyclic Hadamard
test enable us to describe a polynomial ideal that characterizes the set of
cocyclic Hadamard matrices over a fixed finite group $G$ of order $4t$.
Nevertheless, the complexity of the computation of the reduced Gr\"obner basis
of this ideal is $2^{O(t^2)}$, which is excessive even for very small orders.
In order to improve the efficiency of this polynomial method, we take advantage
of some recent results on the inner structure of a cocyclic matrix to describe
an alternative polynomial ideal that also characterizes the mentioned set of
cocyclic Hadamard matrices over $G$. The complexity of the computation
decreases in this way to $2^{O(n)}$, where $n$ is the number of
$G$-coboundaries. Particularly, we design two specific procedures for looking
for $\mathbb{Z}_t \times \mathbb{Z}_2^2$-cocyclic Hadamard matrices and
$D_{4t}$-cocyclic Hadamard matrices, so that larger cocyclic Hadamard matrices
(up to $t \leq 31$) are explicitly obtained.
