No, you don't need to do anything different, really. You are tapping off your speaker level (amplified) right at the speaker without disconnecting it and sending the same signal to the compatable speaker level input which has the necessary matching components to work correctly without damage. The difference between a home audio system as compared to a pinball audio system is mostly what sound information is present in the low frequency ranges. The average subwoofer is going to only "see" frequency information below a maximum of something around 200 hz. There's probably going to be less going on down in that frequency range in a lot of games, as compared to what you might see going on in a commercial recording that you would play on a home audio system. That's the only difference, but the theory of operation is pretty much the same.

I've got a sub hooked up exactly this way on my F-14 Tomcat, which I chose specifically because it has some nice low frequency explosion sounds. You are going to need to keep your existing speakers to reproduce the higher frequency sounds.

You don't need a really thick wire gauge for a short run with the sub close to the game, it would be desirable to increase the wire size for a longer run to reduce loss. The pinball amp is comparably low powered. I use a quality 16 AWG speaker and I think that is overkill.

If you have a lot of cash or spare audio equipment, you could even try this. I think it is kind of a waste to dedicate a whole mixing console, but if you have extra lying around, why not?
http://www.rjbaudio.com/BlackKnight/bkextreme2.html

This speaker has active amplification and is expecting to amplify a line level input. You are sending amplified power into it by clipping to the speaker leads! Discontinue using or there may be permanent damage caused.

Another option is to add an interface to what you have. Google "speaker to line level converter"