[Jeffhs]

This is why just about everyone on VK (myself included) tells anyone about to restore an old TV or radio not to power up the set before recapping it (at very least replacing the power supply filters). Your Admiral TV is 63 years old; many if not most of its original capacitors are defective or are about to fail. Don't plug it in again until those filter caps are replaced with new ones; for the best possible performance, I'd replace the old wax paper caps as well. The job will take some time, but it will be worth it when you see the vastly improved results. When they are operating at peak efficiency, these older TVs can, and often do, outperform many times over today's CRT or flat-panel sets. Admiral was a well-known TV and radio manufacturer, known for quality products, from the '30s through the seventies, so your set was well-made and should serve you well once the caps are replaced, the tuner and controls are cleaned, and any weak or doubtful tubes replaced.

I would say you were extremely fortunate that the power transformer was not damaged or destroyed when the set started smoking (though the rectifier tube may well have been cooked as well, as bandersen mentioned). A short in a filter capacitor will put a direct short circuit across the transformer's B+ windings, ruining the latter in seconds if the set's power supply is not fused.

BTW, do not remove the rectifier tube while the set is plugged in and turned on, and never install the tube with the switch on and the set playing. This could damage or destroy the power transformer or other components of the power supply. I blush to admit I did just that about 40 years ago with a 1951 Majestic b&w roundie and got away with it (the set worked afterward), but I think that was a stroke of very good luck. Usually you do not get second chances with these; once the transformer burns out or shorts you may have a very difficult time finding an exact replacement.

BTW (2): Another thing that can and often does destroy the power transformer in parallel-filament televisions is if the horizontal oscillator stops or is far off frequency. The reason is that very old TVs had horizontal output stages that operated on grid leak bias. If this voltage is missing on the grid of the horizontal output tube for any reason, the plate current of that tube will go off the charts and will overload the power transformer. If the power supply is not fused, the transformer will burn out in very short order.