Here’s what first occurred to me; there may be ways to streamline the process, but this is reasonably quick.
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Duplicate the multiselect field, copying cell data. (Strictly speaking, this is an optional step, as you can convert the original field in place. Me, I like the security of working on a copy of the data.)
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Convert the duplicate column to be of the single line text type. This will result in each cell now containing a concatenated list of tags, separated by commas.
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Create a new column of checkbox type. Check the box for each record needing the new tag.
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Create a new column for formula type. Select the formula to
{original tag string} & IF({needs new tag},",[NewTag]","")
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Duplicate the formula column you just created, copying cell data. (Again, an optional step, as you can convert the column in place.)
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Convert the duplicated formula column to the multiselect type. This will return each tag string to the corresponding grouping of multiselect tags.
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Once you’re happy with the results, delete the original and all intermediate columns, leaving only the updated multiple select field.
One caveat: This will likely break if any of the original tags contains an embedded comma. If that’s the case, as Step 0 edit the field configuration and change the offending tag.
Initially, I created the new tag column as a text field and simply Ctrl-V pasted the tag into the appropriate rows; the subsequent formula merely concatenated original and new tags with a comma separator. The checkbox method, though, requires even fewer keystrokes.
I realize these aren’t the clearest of directions, so I’ve implemented a demo base showing intermediate steps here.