No, there is no observational evidence so far.
We already have some excellent models using quantum effects which would explain a rebound (which might result in a cycle) if future observations uncover signs of one having occurred.
There is no need for amateur theorists, for instance, to offer theoretical ideas to explain hypothetical cycling. Such ideas have already been discussed a whole lot at a professional research journal level---there is more than adequate theoretical preparation.
What is needed now is new instruments sensitive enough to look for features in the ancient light which are predicted by bounce models. It is an area where observation needs to catch up. It would be progress simply to be able to rule out some of the proposed models.
Evidence against would be as welcome and helpful, at this point, as evidence for.
You are asking a question about quantum cosmology (QC) and early universe phenomenology. Phenomenology is the line of research that explores ways of testing theories by observation.
You might find it educational to glance over some of the QC pheno literature. It's a way of contacting the reality of current research in this area and getting some taste of it, or overall impressions. It's not going to be fully understandable, but you can get the flavor and an idea of the main themes. Some article discuss the "observational footprint" of this or that bounce cosmology model. Here are some 46 pheno-oriented papers that have appeared since 2009.