Is there a better promise? Is there any foreshadowing of a Covenant that relates to the Messiah?

The entire story of the Passover, and Exodus mirrors that of the Mashiach. It is a foreshadowing, or a "type" of the Mashiach:

Passover has bondage that represents sin, lack of the law, and oppression.
Israel is led out by G-d as the redeemer.
There is a sacrificial lamb on the door-posts so that those inside were saved and passed over.

The "communion" which was Jesus celebrating the Pessach/Passover ties this togther.

And as they were eating, Jesus took bread, and blessed it, and brake it, and gave it to the disciples, and said, Take, eat; this is my body. And he took the cup, and gave thanks, and gave it to them, saying, Drink ye all of it; For this is my blood of the new testament, which is shed for many for the remission of sins.

Matthew 26:26-29 (KJV)

But Christ being come an high priest of good things to come, by a greater and more perfect tabernacle, not made with hands, that is to say, not of this building; Neither by the blood of goats and calves, but by his own blood he entered in once into the holy place, having obtained eternal redemption for us. ... And for this cause he is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.

Hebrews 9:11-12, 15 (KJV)

And as it is appointed unto men once to die, but after this the judgment: So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation.

Hebrews 9:27-28 (KJV)

In fact there are Midrashim about Moses and the Torah Covenant and how there would be a redeemer!

"Fear not, Abraham; I am thy shield and thine exceeding great reward," refers to the glorious epoch of Messiah. The Patriarch was apprehensive lest the covenant made with him might not prove lasting because of the sins of his descendants. God here gave him the assurance that, though his descendants fall into sin, there shall be one great and noble amongst them, who will be qualified to say to the avenging angel, "Stay thy hand." "Him will I accept and he shall be a pledge for my people."

Midrash on Song of Songs 1

Seeing in his spirit of prophecy that the time would come when the "Mishkan" (the Sanctuary) would cease to exist and the Shechinah dwell no more in Israel's midst, Moses was anxious to know by what means the sins of his people would then be expiated. The Almighty vouchsafed the information that He would choose a righteous man from their midst, and make him a (pledge) for them, and through him their sins would be forgiven.

Midrash Exodus Rabba 35

Not only are there the Midrashim above, but there also notes in the Talmud about the Temple before its destruction
The temple after the time of Jesus's death would no longer be needed. Multiple signs appeared.
In Matthew 27:51 it talks about how at Jesus's death there was an earthquake and the veil in the temple was torn from top to bottom. This can be seen as a sign that the Holy of Holies was now open to all through his atonement.
It is generally recognized that his death would have been around 30AD and that the temple was destroyed by the Romans in 70AD. The forty year period between this time is actually mentioned in the Talmud...

"Forty years before the destruction of the Temple, the western light went out, the crimson thread remained crimson, and the lot for the Lord always came up in the left hand. They would close the gates of the Temple by night and get up in the morning and find them wide open"

Jerusalem Talmud - Jacob Neusner, The Yerushalmi, p.156-157

"Our rabbis taught: During the last forty years before the destruction of the Temple the lot ['For the Lord'] did not come up in the right hand; nor did the crimson-colored strap become white; nor did the western most light shine; and the doors of the Hekel [Temple] would open by themselves"

Babylonian Talmud Yoma 39b