You can maintain the last offset of the read pointer in the file. You can do sth like this
long lastOffset = 0;
using (var fs = new FileStream("myFile.bin", FileMode.Open))
{
fs.Seek(lastOffset, SeekOrigin.Begin);
// Read the file here
// Once file is read, update the lastOffset
lastOffset=fs.Seek(0, SeekOrigin.End);
}