Increase Serial Rx buffer
Use timer interrupts to read and store the data from serial. Lets assume you are receiving data at baud rate of 115200 in default configuration (8data bits, 1 stop bit), 115200/9=12800 bytes per second, approximately 80 micro seconds for every byte, create a timer interrupt that is called once every 2 millisecond(Choose best time for your application making sure that receiver buffer does not overflow), In 2milli seconds 25bytes of data will be available in the buffer. So you can read data in batches of 25 each. From the interrupt read and store the available data from the buffer.