There is no need to paste into Excel to do this. Just make sure there's an (extra) empty paragraph below the table (so that there is at least one empty paragraph between the table and any following text. Then select the last three columns and drag them below the empty paragraph mark.
Before:
After:
You can then split the table between two newspaper-style columns as described.
Word really doesn't like to do that kind of thing. It considers the table to consist of rows; you can split between one row and the next, but not between columns.
If you have Excel available, the easiest method is to copy the table from Word and paste it into a blank Excel worksheet. Then select the first three columns, copy them, and paste them into the Word document after an empty paragraph. Press Enter after the three-column table so there's at least one empty paragraph there, too. Finally, copy the second set of three columns from Excel and paste them into Word after the empty paragraph (that empty paragraph makes sure that Word doesn't combine the first table and second table into one long table).
To place the two tables side by side, select the two tables (and the empty paragraphs above and between them) and click Page Layout > Columns > Two.
Word automatically inserts section breaks before and after the selected material (the one after the second table is invisible in this shot) so the rest of the document remains as one column.
