Do as Fran suggested and use a wider seam allowance and starch the fabric first and press it dry. Handle the fabric delicately, without straining on the seam, as you gently fold it open to press for your second, enclosing seam. After you press it, gently open it to trim the excess, but I would not make it less than 1/4". If you're afraid of cutting your garment also just slip a piece of cardstock or file folder between the seam and bottom layer and cut above the card.
Also be sure to use a smaller needle on the silk chiffon, so you get a smaller hole punched in the fabric. I'd probably also use a thinner thread. I'd use one of my Rayon machine embroidery threads. Someone else will likely have a better idea what size needle and thread to use.
Good luck!
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