This set has two 6V6 audio output tubes in a push-pull configuration. The speaker looks to be a 4-ohm impedance. With a p-p output, you'll need an xfmr with a center-tapped primary. You're trying to match the tubes to the speaker, but I don't think you have to be too precise to obtain good performance.
Both AES and RadioDaze have several xfmrs that are pre-selected for your set-up. The cheapest one I see at RadioDaze is their HX125H, 8W, 10K:2/4/8 ohms, $19.95. AES has the same unit for the same price, their P-T125H. This is a garden-variety Hammond xfmr, which is probably as good or better than the original in your set. If you want to go up to hi-fi rated xfmrs, the sky is the limit, price-wise.
Thomas
If you look in the schematic you'll see two tone condensers going from each primary half to the chassis. One or both of these is shorted, causing B current to flow directly to the chassis. This will eventually blow the transformer, but for now it isn't bad yet. When it stops smoking, then you know that you have blown it. Replace these condensers and it should stop smoking. If these condensers are leaky, it is likely that others are leaky within your radio as well. You should not operate your radio until all condensers have been replaced, or at least all which are leaky. With your resistance meter set to its most sensitive setting (X10,000), you should find no leakage, not even in the millions of ohms. Disconnect one lead whenever you test a condenser. If the condenser has more than two leads (electrolytic, for instance), disconnect all but one lead. If electrolytic, disconnect all except the common negative. If your meter springs up at all, it must fall back to exactly where it started. Normally condensers smaller than .001 MFD will not spring the needle up at all. However, all leaky condensers will spring the needle up, regardless of value, and the needle won't fall back to where it started. If it lingers up even by the most microscopic amount, the condenser is unacceptable. Electrolytics may linger up with a leakage of about 5 or 10 meg. All others must have NO leakage. Small leakage won't cause your smoking problem, but it will ruin operation of the set. It will cause distortion and other malfunctions.
Remove that transformer and test it to see if you've damaged it yet. It is not necessary to remove the transformer from the speaker or chassis (wherever it is mounted). Just disconnect the primary and secondary leads. Check the primary and secondary for continuity. Then replace those tone condensers and check out all of the other condensers within the set. It is extremely poor practice to power up a radio without first testing all of the condensers. Keep in mind that your fingers leak current. Do not touch the meter or condenser leads when checking condensers for leakage.
Thomas
Thomas