If your machine doesn't have a throat plate with a smaller hole, there is a solution. The solution is placing a stabilizer under the trim.
Any one who is familiar with machine embroidery stabilizers knows about stabilizers but rarely think about incorporating stabilizers over into their normal sewing process. If you haven't dabbled in machine embroidery, stabilizers are a whole new world.
I think every one has a stabilizer in their house... one that works fine for a trim or fabric that wants to get dragged under the throat plate. Plain and simple gift wrap type of tissue paper works wonders. Even used gift tissue works. If it is very wrinkled. carefully iron the tissue. Some times you may need multiple layers but you can sew through the tissue and it will tear away when you are done sewing. If some of the tissue paper does remain, just like a tissue left in a pocket disintegrates in the washing machine, so will the tissue stabilizer.
There are products called "temporary spray adhesives", usually found with machine embroidery or quilting supplies, which will assist holding the fabric in place on the stabilizer. This is a spray glue that will work for a limited period of time to hold your fabric in place. A light mist is all that is usually needed to hold the fabric in place. You can also use the temporary spray adhesive to hold multiple layers of the tissue paper together. Always protect the surfaces you are working on to prevent unwanted over spray of the adhesive. In some case, such as trying to apply a decorative stitch to a thin ribbon, using a spray adhesive to hold the ribbon on a sheet of tissue, gives you a bigger area to handle and control the ribbon as it is fed in to the sewing machine.
Stabilizers come in many forms. Some stabilizers can dissolve in water and totally disappear from your work if you use a water soluble stabilizer. You can use a heat away stabilizer. This type of stabilizer vanishes under the heat of an iron. Heat Away stabilizers are my favorite when I am working with a stretchy fabric. You can read more about this in How to Hoop and Embroider a Tee Shirt or Stretchy Fabric
Many people have fancy decorative stitches built in to their sewing machine that they never use. Usually because they are not pleased with the end result of stitching those decorative stitches. Many times the decorative stitches seem all scrunched up. Stabilizers can play a role in improving the quality of decorative stitches that seem to be giving you difficulty. Placing a stabilizer under the fabric helps hold the fabric smoothly while you are sewing.
Many fine or thin fabrics seem to pucker as you sew. So then you start playing with the tension on your sewing machine... until you are ready to toss the whole project. Don't hesitate to place tissue paper under the fabric, or between layers of the fabric. The stitching will sew as if there is a firm fabric and your stitch quality will improve in the process. I find this especially helpful when I am sewing sheer curtains.
Stabilizers can help you achieve joining two different weight fabrics. For example, if you were going to applique a snowman that is going to be a fuzzy fury fabric on a tee shirt weight fabric. If you place stabilizer under the fabrics you help the fabrics to stay evenly stabilized.
Stabilizers can also help you make stitching show in fabric that it would normally be sunken into the fabric fibers. Fur fabric is a prime example. If you place a stabilizer on top of, as well as below, you can "trick" the stitches into staying on top of the furry fabric rather then the stitches disappearing into the fuzzy fur. When you use a stabilizer on top, you must use a stabilizer that will vanish or dissolve when you are finished.