JDriver, I agree with you that you have to get a new medical if Fedex requires it as stated in the question but you are wrong about FAR 61.23.
(d) Duration of a medical certificate. Use the following table to determine duration for each class of medical certificate:
If you hold And on the date of
examination for your most recent medical certificate you were And you are conducting an operation
requiring Then your medical certificate expires, for that operation, at the end of the last day of the
(1) A first-class medical certificate (i) Under age 40 an airline transport pilot certificate 12th month after the month of the date of examination shown on the medical certificate.
(ii) Age 40 or older an airline transport pilot certificate 6th month after the month of the date of examination shown on the medical certificate.
(iii) Any age a commercial pilot certificate or an air traffic control tower operator certificate 12th month after the month of the date of examination shown on the medical certificate.
(iv) Under age 40 a recreational pilot certificate, a private pilot certificate, a flight instructor certificate (when acting as pilot in command or a required pilot flight crewmember in operations other than glider or balloon), a student pilot certificate, or a sport pilot certificate (when not using a U.S. driver's license as medical qualification) 60th month after the month of the date of examination shown on the medical certificate.
(v) Age 40 or older a recreational pilot certificate, a private pilot certificate, a flight instructor certificate (when acting as pilot in command or a required pilot flight crewmember in operations other than glider or balloon), a student pilot certificate, or a sport pilot certificate (when not using a U.S. driver's license as medical qualification) 24th month after the month of the date of examination shown on the medical certificate.
(2) A second-class medical certificate (i) Any age a commercial pilot certificate or an air traffic control tower operator certificate 12th month after the month of the date of examination shown on the medical certificate.
(ii) Under age 40 a recreational pilot certificate, a private pilot certificate, a flight instructor certificate (when acting as pilot in command or a required pilot flight crewmember in operations other than glider or balloon), a student pilot certificate, or a sport pilot certificate (when not using a U.S. driver's license as medical qualification) 60th month after the month of the date of examination shown on the medical certificate.
(iii) Age 40 or older a recreational pilot certificate, a private pilot certificate, a flight instructor certificate (when acting as pilot in command or a required pilot flight crewmember in operations other than glider or balloon), a student pilot certificate, or a sport pilot certificate (when not using a U.S. driver's license as medical qualification) 24th month after the month of the date of examination shown on the medical certificate.
(3) A third-class medical certificate (i) Under age 40 a recreational pilot certificate, a private pilot certificate, a flight instructor certificate (when acting as pilot in command or a required pilot flight crewmember in operations other than glider or balloon), a student pilot certificate, or a sport pilot certificate (when not using a U.S. driver's license as medical qualification) 60th month after the month of the date of examination shown on the medical certificate.
(ii) Age 40 or older a recreational pilot certificate, a private pilot certificate, a flight instructor certificate (when acting as pilot in command or a required pilot flight crewmember in operations other than glider or balloon), a student pilot certificate, or a sport pilot certificate (when not using a U.S. driver's license as medical qualification) 24th month after the month of the date of examination shown on the medical certificate.