The answer to your question is yes. The existence of such expanders which proved by probabilistic method, give us an optimistic view for search to find non-Cayley expanders. But, a first class of such graphs are $k$-regular bipartite which has some parallel edges. So, for your question these are not interesting. There is a general method which you can use for construction non-Cayley expander from an other family of expanders. The only key in this way, is preserving the below bound for the Cheeger constant. For more results and updated things, you can see these two resources, specially the section 3.1 of the second one:
1) Expander graphs, E. Kowalski, ETH Zurich – Spring Semester 2016 Version of February 24, 2016
2) Expander Graphs and Explicit Constructions, Angeliki Lountzi, U.U.D.M. Project Report 2015.