I don't think any such gridded data is available publicly.

You may need to model such data. The paleoclimatic parameters were obtained by applying the Coexistence Approach (CoA) to two Chinese sites. See the attached paper for more detail.

Global warming and rainfall: Lessons from an analysis of Mid-Miocene climate data http://sourcedb.ib.cas.cn/cn/ibthesis/201108/P020110824610615212575.pdf

Regards

Link of another similar article which may help-

Article Middle Miocene climate and vegetation models and their valid...

Regards

I don't think there is a data base but you can find many papers on this topic.

Example: https://www.researchgate.net/publication/324654883_Late_Miocene_climate_cooling_and_intensification_of_southeast_Asian_winter_monsoon

Biswa Bhusana Mahapatra

You could visit timetree (www.timetree.org) for some information such as related articles.

I am also interested in reconstructing the Late Miocene (~8-5 Ma) climate for which I am using, Siwalik micro mammals, fossil tooth enamel as well as pedogenic nodunes.

Article A new genus of treeshrew and other micromammals from the mid...

Conference Paper Stable isotopic (δ13C and δ18O) analysis of late Miocene Siw...

Conference Paper Évolution géomorphologique ancienne pour une vallée du versa...