In a large bowl, whisk to combine the mayonnaise, olive oil, vinegar, mustard, 1/2 tsp salt, and 1/4 tsp black pepper.
Place the cut up red potatoes into a large pot, and fill with enough cold water to cover the potatoes, plus an inch.
Bring the pot to a boil over high heat, and cook the potatoes until fork tender. The time will really vary depending on if you sliced the potatoes or cut them into chunks, and how large, but for me it usually only takes about 5 minutes once the water is boiling, about 10-15 minutes total.
Once the potatoes are fork tender, drain them in a colander and let them sit for a few minutes for the excess steam to come off.
While the potatoes are still warm, toss them well with the dressing.
Stir in the celery, scallions, and chives, then serve warm or chill for several hours before serving. Enjoy!