Differential correction for certain days in the past

Given the availability of data collected at the same time for a base station (i.e., the so called "reference") and a rover station (i.e., the receiver placed on the point whose coordinates need to be computed) it is indeed possible to perform differential positioning and compute precise coordinates.

Normally, the smallest set of data needed in e.g., an open source Global Navigation Satellite Systems (GNSS) processing software like RTKLib includes:

• Raw observations from the reference and rover stations.
• Satellites orbits and clocks, generally included in the "ephemeris" files that can be downloaded from international GNSS analysis centers (e.g., European Space Agency).
• A-priori information on atmospheric conditions (e.g., ionospheric and tropospheric disturbances available in the NASA data repository).

Other ancillary products (e.g., satellite code phase biases, receivers and satellites antenna corrections, earth rotation parameters, input data for ocean loading, ...) might be necessary according to the required accuracy level (i.e., from the few meters to the few millimeters) and are also publicly available.

It depends how you collected the data in the first place. You mention Pathfinder - do you mean Trimble Pathfinder Office? If that's the case, and if you used Trimble TerraSync or similar to collect the data, then you can certainly post-process it. There's a wide range of publicly-available base stations which are available through the Pathfinder Office post-processor tool.

If what you have is NMEA files, I don't know of any way to post-process it.