Ford's Fox chassis was around for over two decades, and when the Mustang was beginning to become popular for hot rodders again in the mid-1980's, the apparent weaknesses of the platform began to show up rather quickly. Not that it's a bad platform; it's rather good and tends to hold up well with time. However, as any motorhead will tell you, more is better. If you can stiffen up a chassis you'll not only get a stronger car, but you'll also get improvements to the ride and handling. Short of a full-out roll cage there are several ways to improve the functionality of the Fox frame. The following are great places to start.
Subframe connectors are excellent for helping to keep your underside together. They connect the points in the unibody "framerail" that are severed in production. Weld-on types are the most effective (while also being race-legal) and should be the only types considered. Bolt-ons will only create elongated holes with time. People have also reported that Mustang units do indeed fit on the Fox Cougar, so you can probably use these as well. Or you can have custom pieces made for relatively cheap; most people use simple box tubing to create their own. After installation, you'll have to paint up all the welds so they don't rust.
Jacking rails work as a combination of outer rocker area braces and subframe connectors in one. While weighing in at around 30 lbs. each, the support that they give will benefit you greatly, especially on cars with a "colorful" history. Plus, you can place a jack under any part of the side of the car, and you're safe to lift up the car. Note that the jacking rails pictured here have the seat cross braces; those are Mustang only and will not work on the Cougar/Thunderbird chassis. You can have custom seat braces made, however.
Long a popular item with Mustang owners, the strut tower brace connects both towers to the cowl, triangulating the engine bay with a stiff brace that keeps cowl flex down. You can use a Mustang brace under the hood of your Cat with minimal problems. Since the washer fluid tube is in a different spot in Mustangs, you may have to relocate yours in your Cougar to allow the brace to bolt into the cowl area. Or you can have a custom made brace. These are mainly for the 5.0 V8 cars, although the brace should clear the 3.8 V6 (both CFI and EFI versions). Braces for the turbo-4 must be custom made.
The G-load brace (aka as the "K"-member brace) resides happily underneath the engine cradle for more underchassis support. In conjunction with a strut tower brace, this bar will keep the front part of the car solidly in place and will significantly reduce chassis flex. Fortunately, we're in luck here: a Mustang-style brace will work. The engine mounting points can vary, but the K-members are virtually identical. There are 2-point and 4-point varieties available in the aftermarket; the 2-point are easier to put in, and are quite strong. The 4-point, on the other hand, are a bear to install, but will be even stronger. It seems that Mustang K-members are a little different than our cars at the mounting point area, and you may need to heat up and bend the G-load brace slightly to fit the Cougar/Thunderbird chassis.
Torque boxes are located just in front of the rear wheels underneath the car. The lower control arms (from the rear axle) bolt into the torque boxes...and therein lies the problem. With added torque to the rear axle, the chassis tends to flex and when too much stress is evident, it has to release its energy somewhere; the torque boxes are usually the first to suffer the consequence. They will tear and separate along the sides and bottom. All Fox torque boxes are about the same, and there are a few Mustang-oriented products that specifically address strengthening torque boxes. This requires removal of the lower control arms, as well as welding, but it's well worth it. Look at it as insurance for the future power you'll be shelling out back there. Even simple welded plates over the bottom of the box will help.
A Panhard bar helps keep the rear axle planted by bolting into the frame/floorpan. The side-to-side motion of torque to the rear axle is severely curtailed by these bars. Currently there are several available for the Mustang. Since the upper control arms from the Cougar and Mustang are different, a Mustang-spec Panhard bar setup may not work. There are universal bars available that should work adequately, available from aftermarket mail-order companies.