If the ship in the previous question is proceeding from the first UK arrival port to another, can the crew member’s engagement be prolonged by the master for any further period?

Yes. 

If after arrival at the first port (in the UK in that particular case) the ship is due to proceed to another port or ports in the same country, then notwithstanding any notice given, the engagement will automatically continue either until arrival at the other port (or the last port in the country if more than one), or for another 7 days from arrival at the first port, whichever occurs first.