I can see two possible solutions for mow. Both of these are not tested.
Use the between operator in toolbar searching. The code for this is bt (set is as default if you want only this possibility). This will be possible only if your date-range picker widget support custom separator between the dates. In jqGrid the separator is … and can’t be configured – it is hard coded for now. Demo with description is here
The second option is to use a custom search field where you should add your own code to separate the dates and build the search string, but I think that this is a little complex.
Please let me know if you have a problem with the first option – we can make this better.
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