Here is another tried and true MJL Warm Up.
It is the word "Kat" repeated on do-mi-re-fa-mi-sol-fa-re-do.
This exercise, like "Kitty Kitty Kat" before it, also involves the front and back of the tongue tongue, the brightness of the “cat” vowel, and adds in a more complicated pattern of ascending and descending thirds.
I have often taught the pattern as 1 3 2 4 3 5 4 2 1 to students starting out, which greatly aids developing their ear; then I replace the numbers with the word to exercise the technical aspect I am after.
I hope you enjoy!