# [Charcoal], <s>43</s> 35 bytes

    ＷＳ⊞υＬι§ΦＥ⊕⌈υ⁺Ｅι⁺ιＬΦυ›νμυ⬤ι⁼λＬΦι›νμ⁰

[Try it online!][TIO-ku897n2e] Link is to verbose version of code. I/O is as a Young Diagram using `-`s as the fill character. Explanation:

    ＷＳ⊞υＬι

Input the diagram.

    Ｅ⊕⌈υ

For each possible number of rows from `0` to the length of the first row (inclusive)...

    ⁺Ｅι⁺ιＬΦυ›νμυ

... calculate the transpose of that number of columns, concatenating with the original input.

    §Φ...⬤ι⁼λＬΦι›νμ⁰

Print the first symmetric result.

Example: At each stage, an `i`-by-`i` square is prefixed to the original input (marked by `#` signs) and then extended by transposition (marked by `=` signs). The first symmetric result is then the one with `3` prefixed rows.

                                            #####===
                                    ####=== #####==
                            ###===  ####==  #####==
                    ##===   ###==   ####==  #####==
            #===    ##==    ###==   ####==  #####=
    -----   =----   ==---   ===--   ====-   =====
    ----    =---    ==--    ===-    ====    ====
    -       =       =       =       =       =
    0       1       2       3       4       5

[Charcoal]: https://github.com/somebody1234/Charcoal
[TIO-ku897n2e]: https://tio.run/##bY7BCsIwDIbve4oeU@jAu6cdVAYKA5@gbGENpHV27dzb13YOD@J/CH/I/yXpjfb9Q3NKL0OMAlo3xXAPntwIUoouzgaiEld0YzBAUh6rLg8DNKF1A65wJg7o4aanzPYeLbqAQ@5XstFClFKJjuO8JWj39N244/nExaMu1ilhZZESG9wwl/zpGTXPwL8k/SEzdcifplQXVZ9SpXrhNw "Charcoal – Try It Online"