Megan, congrats with the new Eames Molded Plywood Lounge chair. We completely agree with your assessment about the Ikea coffee table being a poor match with the lower profile of the chair. Whether you go round or square seems less of an issue than the point of picking a coffee table that sits at a height compatible with your seating. We also see an opportunity to bring in a bit more colour like you have with the addition of a walnut chair, as the clear acrylic table may result in a room filled with too similar of a palette since the table would give a view of the neutral tone rug below.Ideally, you would consider saving up for something like a Saarinen Low Oval Coffee Table (here's one on the cheap, but that needs work) with a walnut veneer or maybe something square-minimal like this Mott Modern Coffee Table which sits low while providing a sufficient surface area to span the space between your sofa and your chair. Of course, with some time and patience, perhaps you could find a vintage piece, like this wonderful vintage coffee table from Swank in St. Paul, shown here.
And I'm sure many of our readers have their own opinions of tables that would match your decor and layout (and remember, you can always move things around too). Readers...any other suggestions?