I am looking for interesting examples of categories admitting multiple *monoidally inequivalent* [closed](https://ncatlab.org/nlab/show/closed+monoidal+category) (or [compact closed](https://en.wikipedia.org/wiki/Compact_closed_category)) symmetric monoidal structures. 
We know how to construct disconnected toy models [1], but none of direct practical interest. I also seem to recall from [2] that the category of $G$-sets admits two inequivalent such structures, but I am curious to know whether this is a sporadic occurrence or whether there is a large family of interesting examples lurking somewhere out there.


[1]: see e.g. <https://arxiv.org/abs/1803.00708>


[2]: Borceux, Francis. *Handbook of Categorical Algebra*



Inequivalent compact closed symmetric monoidal structures on the same category
A pretty interesting class of examples comes about by classifying compact monoidal *groupoids*. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G \times G \times G \to M$, one can manufacture a compact monoidal groupoid whose category of objects is $G$, whose morphisms are ordered pairs $(g, m) \in G \times M$ where we define $\text{dom}(g, m) = \text{cod}(g, m) = g$ and where endomorphism composition is defined by addition in $M$, and where we define the tensor product by $(g, m) \otimes (h, n) = (g h, m + g n)$. It's the 3-cocycle that furnishes the associativity data, and we get monoidally inequivalent groupoids whenever the 3-cocycles are not cohomologous. This was observed by Joyal and Street in their paper [Braided Monoidal Categories](http://web.science.mq.edu.au/~street/JS1.pdf). 


Now you were asking about the symmetric monoidal case (where we now assume $G$ is abelian). These are also known as *Picard groupoids*. In the simplified scenario where we demand strict associativies and consider only the case where $G$ acts trivially on $M$, in which case the underlying category becomes the product $K G \times B M$ of the evident discrete monoidal category $K G$ with the evident one-object category $B M$, any symmetric bilinear pairing $G \otimes G \to M$ can be used to manufacture a symmetry isomorphism for a symmetric monoidal structure on $K G \times B M$, and these examples are generally symmetric-monoidally inequivalent. I can't tell how far away such examples are from the "toy"examples" you have in mind, but Picard groupoids are surely of interest -- see for example applications to 2-stage Postnikov systems of spectra [here](http://www.math.uchicago.edu/~may/MISC/Picard.pdf). 


