Obviously, it is necessary to get some support for that addition. A complete support wall would be ideal, but if you want to avoid having to make two small rooms with the addition of the support wall, there is a solution. If you look in your basement there should be two or more columns that run in line with the support wall above. These columns, together with the beams they support, are doing the same job as a support wall. If you extended those columns from the floor to the ceiling below the addition and install a beam across the ceiling in line with the columns then you will have the support you need. The room will not be completely open, but with some interior design, it can maintain its open feeling.