If you decide to change the trans lube, use Redline 75W90NS. This will improve things a bit. The reason that a TR gearbox must "warm up" for a better 2nd gear, is because the slider shaft goes into a blind hole, that is, a hole without an oil outlet. So when the oil is cold it's also thicker, making 2nd gear a bit more difficult. Once the oil is warm, 2nd gear is much easier. But I have notice a couple of cars that shifted great into 2nd gear even when cold, but the majority of TRs have the "skip 2nd till warm" issue. Another issue on the TR is the shifter adjustment. When in neutral, the shifter must be "centered" within the shifter gate, both side-to-side and front-to-back. If the shifter is not centered, then an adjustement is needed. Adjusting the shifter on a TR take a bit of patience, and maybe 2 or 3 tries to get perfect. But it's well worth the effort.
Driving the TR and getting smooth takes a lot of practice. Matching revs from one gear to the next can be done by "ear" and once you get the hang of it - believe me it will be smooth. The TR will never shift like a ZF trans, or even like a good Muncie. Another factor most do not consider is, that the TR's shifter is adjustable forward/backwards and also rotationally. Under hard acceleration, the flat-12 motor torque is lifting the left side and pushing down on the right side, effectively rotating the motor slightly - and changing the way the shifter will operate at that instant. You definately cannot "bang the gears" in a TR, but then again the car wasn't intended to be a drag racer.
On my first TR it took me a few weeks of serious practice to get smooth, but once I did it was just wonderful and effortless. Oh how I miss my TR!