Use the full-blown run of the XRD and not only the 10 Angstoem reflections, use chemical treatments such as those for testing the swelling behavior so as to see smectite interlayers and determine the crystallinity and Si/Al disorder of these phyllosilicates.

HGD

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The type of clay minerals and illite/mica may be assessed by looking in particular at the relative peak heights of the 00l series, presence of a peak at 4.49 Å (020) or precise position of the 060 peak (around 1.5 Å). As Harald G. Dill mentioned you will need to run extended traces, which is required for evaluation of the 060 peak. Usually reliable identification is only possible in samples with rather high mica content, or you will need to concentrate the mica before running the samples (e.g. separate the clay-sized fraction or employ some other concentration method). Other minerals present in the sample may have overlapping peaks, so you will need to do a complete mineral analysis of the XRD trace. Brindley and Brown, 1980 (Crystal structures of clay minerals and their X-ray identification) is a good start for more information on XRD analysis (see e.g. chapter 1 – table 1.15 and chapter 2 – table 2.10). Dixon and Weed (Minerals in Soil Environments) also has extensive information on XRD analysis and other methods for identification of (clay) minerals (e.g. chapter Micas). You can also find more specialised literature.

Brindley and Brown can be freely downloaded from the internet, e.g. https://www.academia.edu/51367964/Crystal_structures_of_clay_minerals_and_their_X_ray_identification or perhaps some other website.