The paper wrapper on that transformer doesn't look like it was smoking. If it did, it's more likely something connected to one of the outputs was shorted and not a fault of the transformer itself.
I know this might seem risky to some, but I would check that none of the lugs are bent and touching another, connect power to the transformer and measure the AC output voltages and go from there.
Again a smoking transformer is indicative of a short circuit downstream in the game - most likely lamp based due to high currents and it was fused with a nail..