In answering this question, which is based on the account in John's Gospel, it is important not to read into this Gospel any facts or notions found elsewhere. John's Gospel differs from the synoptic gospels in that it does not mention the Last Supper, instead having Jesus wash the feet of the disciples after what appears to have been the normal supper meal.
Clearly, the author of John's Gospel was aware that the crucifixions could not take place on the Day of the Passover and so made a number of changes from the synoptic account, one of which was to place the crucifixion of Jesus on the day before the Passover. This is evidenced not only in John 18:28 and John 19:31, but even more clearly in John 19:14:
John 19:14: And it was the preparation of the passover, and about the sixth hour: and he saith unto the Jews, Behold your King!
Because the Last Supper is not mentioned in John's Gospel, we need to go to the synoptic gospels to find out more and to answer the question of whether this was taken as the Passover feast.
Mark 14:12 tells us that on Jesus' last full day of freedom, the disciples talked to him about preparation for the Passover feast. This was the day they killed the paschal lamb, so clearly the Passover feast would be celebrated that evening, as soon as the Day of the Passover began at nightfall:
Mark 14:12: 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?
Mark 14:14-18 talk about the preparation for the Passover feast that evening and how Jesus sat down with the disciples for this feast. It was the beginning of the Day of the Passover:
Mark 14:14-18: 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? And he will shew you a large upper room furnished and prepared: there make ready for us. And his disciples went forth, and came into the city, and found as he had said unto them: and they made ready the passover. And in the evening he cometh with the twelve. And as they sat and did eat...
Conclusion
In the synoptic gospels, where the Last Supper is specifically mentioned, it is undoubtedly the Passover feast. John's Gospel does not mention the Last Supper and is sufficiently different in detail and outline that we can not make a judgement on the synoptic account on the basis of John's version.
Addendum: Church Fathers
Origen talks of the crucifixion and the Passover in terms that can only mean he believed that Jesus was sacrificed on the Day of the Passover:
Contra Celsum 8:22: If it be objected to us on this subject that we ourselves are accustomed to observe certain days, as for example the Lord's day, the Preparation, the Passover, or Pentecost, I have to answer, that to the perfect Christian, who is ever in his thoughts, words, and deeds serving his natural Lord, God the Word, all his days are the Lord's, and he is always keeping the Lord's day. He also who is unceasingly preparing himself for the true life, and abstaining from the pleasures of this life which lead astray so many—who is not indulging the lust of the flesh, but keeping under his body, and bringing it into subjection,— such a one is always keeping Preparation-day. Again, he who considers that Christ our Passover was sacrificed for us, and that it is his duty to keep the feast by eating of the flesh of the Word, never ceases to keep the paschal feast; for the pascha means a passover, and he is ever striving in all his thoughts, words, and deeds, to pass over from the things of this life to God, and is hastening towards the city of God. And, finally, he who can truly say, We are risen with Christ, and He has exalted us, and made us to sit with Him in heavenly places in Christ, is always living in the season of Pentecost; and most of all, when going up to the upper chamber, like the apostles of Jesus, he gives himself to supplication and prayer, that he may become worthy of receiving the mighty wind rushing from heaven, which is powerful to destroy sin and its fruits among men, and worthy of having some share of the tongue of fire which God sends.