1. Question Restatement:
Is there a hint in any of Evangelions why Jesus was crucified so close to Sabbath?
This is a very contested, topic - but I hope I have identified all of the potential explanations:
2. Quick Answer - Why the Urgency?
Perhaps the best answer is simply, "because everyone was finally in agreement"?
Matthew 26:18 - And He said, “Go into the city to a certain man, and say to him, ‘The Teacher says, “My time is near; I am to keep the Passover at your house with My disciples.”’”
Pilate's Sense of Urgency:
NASB, Mark 15:15 - Wishing to satisfy the crowd, Pilate released Barabbas for them, and after having Jesus scourged, he handed Him over to be crucified.
From Jewish Leadership Point of View:
NASB, John 11:48 - If we let Him go on like this, all men will believe in Him, and the Romans will come and take away both our place and our nation.”
3. Answer - The High Priest's Sense of Urgency, the Passover:
NASB, John 18:12 - So the Roman cohort and the commander and the officers of the Jews, arrested Jesus and bound Him, 13 and led Him to Annas first; for he was father-in-law of Caiaphas, who was high priest that year. 14 Now Caiaphas was the one who had advised the Jews that it was expedient for one man to die on behalf of the people.
This is the ONLY hint / explanation given in the Gospels - about the Underlying Reason.
The urgency was to fulfill prophecy, and to ensure that Jesus' death coincided with the Passover Sacrifice.
There were TWO Sabbaths the week Jesus died, the Passover Sabbath - the "High Day" - and the Seventh Day Sabbath. See "High Holy Day Sabbaths, Wikipedia".
NASB, John 19:31: Then the Jews, because it was the day of preparation, so that the bodies would not remain on the cross on the Sabbath (for THAT Sabbath was a HIGH DAY), asked Pilate that their legs might be broken, and that they might be taken away.
Matthew 27:25 - And all the people said, “His blood shall be on us and on our children!”
NASB, 2 Chron. 35:11 - They slaughtered the Passover animals, and while the priests sprinkled the blood received from their hand, the Levites skinned them.
Isaiah 52:13 - Behold, My servant will prosper, He will be high and lifted up and greatly exalted. 14 Just as many were astonished at you, My people, So His appearance was marred more than any man And His form more than the sons of men. 15 Thus He will sprinkle many nations, Kings will shut their mouths on account of Him; For what had not been told them they will see, And what they had not heard they will understand.
NASB, John 11:49 - But one of them, Caiaphas, who was high priest that year, said to them, “You know nothing at all, 50 nor do you take into account that it is expedient for you that one man die for the people, and that the whole nation not perish.” 51 Now he did not say this on his own initiative, but being high priest that year, he prophesied that Jesus was going to die for the nation, 52 and not for the nation only, but in order that He might also gather together into one the children of God who are scattered abroad.
53 So from that day on they planned together to kill Him.
The Question was Asking for a HINT of why it was rushed, and why it had to coincide with THAT Sabbath, that "High Day".
Regardless if this explanation is right or wrong - it is the explanation offered in the Gospels - the only explanation.