I agree with the previous advice. Paper & electrolytic capacitors should be
replaced eventually, so why not start now? This article has basic advice
about recapping.

http://antiqueradio.org/recap.htm

You can download the schematic & service data from the BAMA site:

ftp://bama.sbc.edu/downloads/hallicra/

I don't know offhand if all of the SX-100 tubes are glass, in which case you
can look for duds by simply looking to see whether tube lights up inside. I
believe this set uses a 0A2 regulator tube, which will glow purple rather
than the usual orange-ish. Don't freak out -- that color is normal.

Knowing that a tube lights up doesn't tell the whole story, since the tube
might still be too weak to perform well. On the other hand, if you find a
dud or two, you've learned something useful.

You can also check for duds by measuring the resistance of each tube's
filament with an ohmmeter. You can look up each tube's basing diagram at
Nostalgia Air:

http://www.nostalgiaair.org/

Click on Tubes, then enter the type (for example, 6SN7) of each tube you
want to test.

The tube filament is shown in the diagram as little mountain-peak sort of
symbol. In a 6SN7 tube, for example, the filament pins are 7 and 8. If the
filament is good, you should measure very low resistance between those pins
using an ohmmeter. If you measure infinite resistance, the filament has
burned out and the tube is no good.

When you say that the radio "hums," if that means a loud, low-pitched hum
which doesn't change as you turn the volume up or down, then you have bad
electrolytic capacitors in the power supply. Replace those before doing
anything else. You should do that anyway, to avoid burning up your power
transformer if one of them fails.

If by "hum" you mean just the normal faint background hum from a tube radio,
then don't worry about it.

There is a very simple test to tell you whether your audio amplifier is
working. Turn up the volume, then briefly touch a turned-on soldering iron
to the input terminal of the volume control potentiometer. (That is, the
side that is not connected to ground.) If your audio section is working,
you'll hear a loud growl from the speaker.

Voltage checks are tedious, but can be very informative. When making any
measurements on a powered-up chassis, follow the old-timer's rule of keeping
one hand in your pocket when doing the test. This prevents you from getting
a shock by touching two hands to places they should not touch (Bzzzt!).

Have fun. That should be a great receiver when you get it working properly.

Regards,

Phil Nelson
Phil's Old Radios
http://antiqueradio.org/index.html