Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}_p^2$. The title is Integral Duality and the Structure of Tate Cohomology Rings.

Also, there is an unpublished/unfinished paper of Weiss, found here: http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf , which computes $\hat{H}^*(\mathbb{Z}_2^r,\mathbb{Z}_2)$.

Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}_p^2$. The title is Integral Duality and the Structure of Tate Cohomology Rings.

Also, there is an unpublished/unfinished paper of Weiss, found here: http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf , which computes $\hat{H}^*(\mathbb{Z}_2^r,\mathbb{Z}_2)$.

And just to add to your current list (periodic groups): $\hat{H}^*(S_3,\mathbb{Z}_3)=\Lambda[x]\otimes\mathbb{Z}_3[z,z^{-1}]$ with $|x|=3$ and $|z|=4$.

Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}_p^2$.

The title is Integral Duality and the Structure of Tate Cohomology Rings.

Antonio Bellezza's PhD thesis (Pisa, 2002) computes the ring structure of the Tate cohomology for $\mathbb{Z}/p^a\times\mathbb{Z}/p^b$, and also the mod-p Tate cohomology of $\mathbb{Z}_p^2$. The title is Integral Duality and the Structure of Tate Cohomology Rings.

Also, there is an unpublished/unfinished paper of Weiss, found here: http://www.math.uwo.ca/~schebolu/research/Jan/tateprop.pdf , which computes $\hat{H}^*(\mathbb{Z}_2^r,\mathbb{Z}_2)$.