LEM,
On some Boeings, the FMC IS so clever to provide Max Cont thrust following engine failure, but unfortunately not the B737. The Flight Director still provides the proper engine out guidance.
On later A/T computers, the hold mode starts at 84 knots and the A/T re-arms at the airline default height (may be higher than 400 RA).
After flap retraction, the other pilot selects CON on the N1 page which moves the bug to Max Cont thrust (until another mode is selected or engaged). The thrust must be manually set. Disconnect the A/T and select level change for further climb.
(The thrust may be left at max N1 for up to 10 minutes for terrain clearance, otherwise 5 minutes).