Rainbow comes in 7 colors (ROYGBIV -red, orange, yellow, green , blue, indigo, violet). I believe the reason why God used the rainbow as a remembrance not to flood the whole world again because the 7 colors remind God of the 7 days in which he took to create the earth and all that is in it. Besides, rainbow almost always occur after the rain.
Genesis 9:12 And God said, "This is the sign of the covenant I am making between me and you and every living creature with you, a covenant for all generations to come: 13 I have set my rainbow in the clouds, and it will be the sign of the covenant between me and the earth. 14 Whenever I bring clouds over the earth and the rainbow appears in the clouds, 15 I will remember my covenant between me and you and all living creatures of every kind. Never again will the waters become a flood to destroy all life. 16 Whenever the rainbow appears in the clouds, I will see it and remember the everlasting covenant between God and all living creatures of every kind on the earth."
The other reason could be that he created all the land animals including humans on the 6th day before he rested on the 7th day. The land animals and humans were the ones that God totally destroyed during the Great Flood. The 6th day was also the day he blessed all living creatures and humans before He rested on the 7th:
Genesis 1:24 And God said, "Let the land produce living creatures according to their kinds: livestock, creatures that move along the ground, and wild animals, each according to its kind." And it was so. 25 God made the wild animals according to their kinds, the livestock according to their kinds, and all the creatures that move along the ground according to their kinds. And God saw that it was good.
26 Then God said, "Let us make man in our image, in our likeness, and let them rule over the fish of the sea and the birds of the air, over the livestock, over all the earth, [b] and over all the creatures that move along the ground."
27 So God created man in his own image,
in the image of God he created him;
male and female he created them.
28 God blessed them and said to them, "Be fruitful and increase in number; fill the earth and subdue it. Rule over the fish of the sea and the birds of the air and over every living creature that moves on the ground."
29 Then God said, "I give you every seed-bearing plant on the face of the whole earth and every tree that has fruit with seed in it. They will be yours for food. 30 And to all the beasts of the earth and all the birds of the air and all the creatures that move on the ground—everything that has the breath of life in it—I give every green plant for food." And it was so.
31 God saw all that he had made, and it was very good. And there was evening, and there was morning—the sixth day.