You say that the oscillation doesn't go away even with the volume control turned down. This sounds like it is in the amplifier. Again, keep speaker wires away from tubes (except output tube and rectifier, if present), and be sure that grid wires are well shielded/dressed. Also, if you are using the original electrolytics, be sure that they have a low power factor. Otherwise they won't filter high frequency fluctuations in the plate current. A .05 MFD capacitor paralleling one of the main filter capacitors may help this. If you are using new electrolytics, this might not be necessary.
If the amplifier is just poorly designed, oscillation might be dampened by installing tone condensers in various places. Usually one in the plate circuit of the output tube does the trick, though not always.
T.
The best solution is to properly route wires and shield the amplifier, but if you cannot remove the oscillation by these methods, then small value shunting capacitors in the audio circuit might help remove unwanted high frequency oscillation.
Also, if you do have a schematic at hand, note the radio's wiring as compared to the schematic. Make sure that everything is connected properly and measures properly (such as the low end of the volume control). Make sure that grid wiring and resistors are connected properly, and that the resistors/capacitors/wires are not open circuited.
T.