Charcoal, 42 35 bytes

ＡＥθＳθＷ⊟θ«⪫θ¶ＡＥι⮌⪫Ｅθ§μλωθ⊞υι↙←⮌⪫υωＤ⎚

Try it online! Link is to verbose version of code. Edit: Saved 7 bytes mostly by switching from character arrays to strings. Explanation:

ＡＥθＳθ

Read the input square as an array of strings into the variable q.

Ｗ⊟θ«

While the last string in the array is not empty, remove it.

⪫θ¶

Print the rest of the array.

ＡＥι⮌⪫Ｅθ§μλωθ

Rotate the rest of the array by looping through each character of the last string and joining the lth character of every remaining string in the reversed array.

⊞υι↙←⮌⪫υω

Append the previously removed last string to u, which holds the unfurled value, and print it.

Ｄ⎚

Output the result and then clear the canvas ready for the next iteration.

Note that this version outputs the final unfurl on a separate line, if this is undesirable then for 38 bytes:

ＡＥθＳθＷ⊟θ«⊞υι←Ｅ⁺⟦⪫υω⟧⮌θ⮌κＡＥι⮌⪫Ｅθ§μλωθＤ⎚

Try it online! Link is to verbose version of code. Explanation: ←Ｅ⁺⟦⪫υω⟧⮌θ⮌κ reverses the current array, prepends the unfurled line, then reverses the characters in each line, then prints everything upside-down, thus producing the desired result.