Cook pasta according to package directions in a large pot of salted water. Drain the pasta and run it under water. Drizzle with the olive oil and toss well.
Put the pasta in a large serving bowl and stir in the pesto and Italian dressing. Add the spinach, chicken, tomatoes, and feta cheese. Toss to combine. Season the pasta salad with salt and pepper, to taste.
Cover the pasta and refrigerate. If the pasta seems dry before serving, you can stir in another tablespoon of olive oil.