Is is possible to examine migration patterns by county level or smaller?

Unfortunately, state is the lowest level of detail collected by the Census for place of birth (BPL). If it is possible to identify the particular county in which you are interested (see Excel spreadsheet of identifiable counties here), then you could still estimate the in-migration from other states to this county. However, you would not be able to estimate in-migration from other areas in the same state to this county or any out-migration from this county.

While not relevant to birthplace-based migration, IPUMS-USA does offer migration variables that report a respondent’s residence 1 year ago. These variables are available at the PUMA level, which is smaller than the state but might not be an exact match to your county of interest. Please refer to these previous answers for more background on the migration variables: here and here.

Sorry that I could not deliver better news.