If I understand correctly, one easy way to construct the arbitrary signature $\sigma(M)$ of 4-manifold $M$ is by connecting sum the $\mathbf{CP}^2$ manifolds, since $\sigma(\mathbf{CP}^2)=1$.
$$\begin{array}{|c|c|c|c|c|} \hline \sigma(M)& -2 & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline M & \overline{(\mathbf{CP}^2)^{\#2}} & \overline{\mathbf{CP}^2} & S^p \times S^q \times S^m \times \dots & \mathbf{CP}^2 & \mathbf{CP}^2 \# \mathbf{CP}^2 & (\mathbf{CP}^2)^{\#3} & (\mathbf{CP}^2)^{\#4} & (\mathbf{CP}^2)^{\#5} & & \\ \hline \end{array}$$
$$\begin{array}{|c|c|c|c|c|} \hline \sigma(M)& 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline M & \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \hline \end{array}$$
I also understand that the product spaces of spheres have $$\sigma(S^p \times S^q \times S^m \times \dots)=0.$$
What are other ways to construct $M$ with $\sigma(M)$ for arbitrary integers?
I am looking for explicit examples of manifolds with different signatures.
All spin 4-manifolds have $\sigma(M) =0$ (mod 16). So we are looking into some non-spin 4-manifolds.