I was reading Mark 14 again this morning and once again think on the proper ordering of the days of these events and Passover.
Perhaps the following account sheds some light on the subject.
If this analysis is correct it explains the timing, though there are similar aspects of the bread and wine and hymn singing that do occur in the actual passover Seder but it is possible that these facts refer to the supper celebration as described.
Mark 14 King James Version (KJV)
1 After two days was the feast of the passover, and of unleavened bread: and the chief priests and the scribes sought how they might take him by craft, and put him to death.
12 And the first day of unleavened bread, when they killed the passover, his disciples said unto him, Where wilt thou that we go and prepare that thou mayest eat the passover?
13 And he sendeth forth two of his disciples, and saith unto them, Go ye into the city, and there shall meet you a man bearing a pitcher of water: follow him.
14 And wheresoever he shall go in, say ye to the goodman of the house, The Master saith, Where is the guestchamber, where I shall eat the passover with my disciples?
15 And he will shew you a large upper room furnished and prepared: there make ready for us.
16 And his disciples went forth, and came into the city, and found as he had said unto them: and they made ready the passover.
17 And in the evening he cometh with the twelve.
18 And as they sat and did eat, Jesus said, Verily I say unto you, One of you which eateth with me shall betray me.
19 And they began to be sorrowful, and to say unto him one by one, Is it I? and another said, Is it I?
20 And he answered and said unto them, It is one of the twelve, that dippeth with me in the dish.
21 The Son of man indeed goeth, as it is written of him: but woe to that man by whom the Son of man is betrayed! good were it for that man if he had never been born.
22 And as they did eat, Jesus took bread, and blessed, and brake it, and gave to them, and said, Take, eat: this is my body.
23 And he took the cup, and when he had given thanks, he gave it to them: and they all drank of it.
24 And he said unto them, This is my blood of the new testament, which is shed for many.
25 Verily I say unto you, I will drink no more of the fruit of the vine, until that day that I drink it new in the kingdom of God.
26 And when they had sung an hymn, they went out into the mount of Olives.