Insert to particular location in Oracle DB table? Suppose I have a table containing the following data:

Name | Things
-------------
Foo  |   5
Bar  |   3
Baz  |   8


If I want to insert a row, so that the final state of the table is:

Name | Things
-------------
Foo  |   5
Qux  |   6
Bar  |   3
Baz  |   8


Is this possible?

I understand we don't typically rely on the order of rows in a table, but I've inherited some code that does precisely that. If I can insert to a location, I can avoid a significant refactor.
The code you have inherited is broken - and it always has been broken. That re-factoring you'd like to avoid needs to be done. There is no alternative to an explicit order by to guarantee the sort order of a result set and there never has been.

Who knows if the code you inherited always returned rows in the order the original developer 'expected' or not.