I think that most national midweek games are either on NBC Sports Network or NHL-TV. Not sure about every system, but they likely require a specialized sports tier beyond the basic package that would include ESPN and the YES Network.
There is a also a higher-level package (NHL Center Ice?) that enables someone to watch every single out-of-town game. With the season wrapping up, you might want to call and see if they'll give you a free or discounted month as a teaser.
I can't say for sure this year, but in years past I know the NHL Center Ice (now called NHL.TV) package price went down after a certain point in the season. At this point, the Devils have less than 10 games left in regular season. I assume the Leafs are in the much the same position. When the playoffs start, many (but perhaps not all) of those games will be available on the various NBC channels and will be blacked out on the NHL.TV package. Take all this into consideration when shopping for the package.
Also note, if you have a streaming device you can subscribe to NHL.TV via the NHL app. No cable subscription needed. (But again, games broadcast on the NBC channels will be blacked out.)
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So, we do not currently have cable but, for various reasons, we are thinking about getting a basic package. My kid is pretty into hockey, particularly the Toronto Maple Leafs. If we want to watch their games do we need to buy a special package or will they (mostly) be available via regular cable?