The answer to your question is a lot more complicated that you'll probably like.
First, it's true that the 4 ohm cabinet will be drawing more power than the 8 ohm cabinet. And no, there's no practical way to rewire the 8 ohm cabinet for 4 ohms. But drawing more power doesn't necessarily mean it will be louder. To conmplicate things further, it might well be louder, in that it's putting out more acoustic energy, but not sound as loud due to the energy being in a part of the spectrum where our ears are not very sensitive. No way to tell without more information.
If the 4 ohm 4x10 is similar to the 8 ohm cabinet you presently have, then the difference in volume is not likely to be audible. However, a second 8 ohm 4x10 similar to the one you already have most likely would be.
Things affecting loudness include (but are not necessarily limited to) sensitivity, that is, how much sound the speaker puts out for a given input; Vmax, which is the maximum amount of air the speaker can move at full travel; power handling; the type of cabinet the drivers are installed in; power compression; the linearity of the magnetic field in the voice coil gap.
You say it's not loud enough. Not loud enough for what? "Not as loud as my other rig" isn't a very good answer, because we don't know how loud your other rig is, nor what you're using them for, so please be specific as to why you need so much volume. 250 watts or so into one 18" and a 4x10" should be quite enough for most applications, and if you need more, well, that's what sound reinforcement systems are for. Your situation may be different, though.
"A 350W, 4 ohm amp running at about 5" doesn't tell us much, either. It's quite possible that, even with the master on 5, that the 350W amp is already putting out all it has, and may be well past its rated output. What kind of amp is it, and what speakers is it running into? For that matter, what are the 1x18" and the 4x10"?
Based on the little bit of information I have, here's my guess: the 350W amp has a response curve that cuts off a lot of the low bass, leaving more power available for the low mids and mids, giving it more output in the range our ears are most sensitive to, so it sounds louder. The technical term for that is that it has a greater apparent volume. You can do something similar with the Coliseum: cut the low bass, and leave the other tone controls on 5. Then turn up the volume. It's apparent volume should be much greater than before. Once it's loud enough (assuming it gets there), you can add in some low bass until you like the sound or it starts to distort more than you like. Of course, this might not be necessary. You might like the sound you get with the low bass turned down. You can also adjust the tone using the other tone controls to make it more to your liking. However, IIRC, the Coliseum tone controls are set up like an equalizer, essentially a separate volume control for each frequency band - like a graphic equalizer, but with rotary controls. If that is so, then you'll likely get your maximum volume with the controls set flat, with the exception of the low bass. Low bass uses a lot of power.
One other thing, which could make everything else moot. Are your two cabinets the same polarity? If you put a 9V battery on the input to the cabinets, do they move the same direction? If not, they'll fight each other, kill the volume, especially in the lower frequencies.
