Yes some Au Jus will definitely push it over the top. Just put a pan under the meat while it cooks & use the drippings as a base for the Au jus. I usually start with some carrots, onions, garlic, celery, & a can on French onion soup in the pan, & let them cook down as the meat juices drip in the pan as well. Then all we do is strain out all the veggies & put it in a pan with some red wine & boil it on the stove, then reduce to a simmer for about 15 minutes. It will be done when you are done resting the TT. I rest mine on the counter with a foil cover. If you put it in a cooler with towels it will continue to cook & your med rare will turn into well done. I would pull it at 130 & after a short rest on the counter the carryover cooking will bring it up to 138-140.
Here is an Au Jus recipe from Chef JimmyJ, it's a good one.
ChefJimmy’s Au Jus
1- Lg Onion,
4-5 Carrots,
3-4 Ribs Celery
3-4 Peeled Cloves of Garlic
Toss them in a pan under the Beef, and let the whole deal Smoke for one hour,
THEN add 4-6 Cups Beef Broth,
2 Tbs Tomato Paste,
1/2tsp Dry Thyme (4-5 sprigs Fresh)
1-2 ea Bayleaf
Finish the Smoking process to the IT you want.
While the Roast is resting, dump the pan juices veggies and all into a 2-3Qt Sauce pot and add 1Cup Red Wine, something you like to drink, and bring the Jus to a boil, lower the heat and simmer 20-30 minutes. Strain out the veggies and let the Jus rest a minute or so for the Fat to rise. Skim off the bulk of the fat then using strips of paper towel laid on top of the Jus then quickly removed, take off the last little bit of fat.
The purpose of Smoking the Vegetable for 1 hour before adding the Broth and Herbs is...The Smoked vegetables Roast in the Dry heat concentrating their Flavors and Sweetness giving the finished Jus a Richer, Deeper, Full Flavor.
Serve the sliced Beef Au Jus or thicken the Jus to make Gravy
Hope this helps!
Al