I am a hockey photographer for a blog and I have a Canon T5 with the standard lens and the telephoto EF-S 55-250mm image stabilizer. I want to purchase a new lens to take more close up shots and across the ice net to net, what type of lens should I be buying? I also don't want to go into a huge expense since my photography is volunteer.
To me, your options fall along a spectrum.
At one end are optics that will perform well at producing the shots mentioned in the question. I can't say what a 'huge expense' is for you, but the lenses that will simplify getting those shots are bigger and faster pieces of glass that start around 2x the cost of your existing camera and lens and then more or less keep doubling in cost from there.
At the other end is learning how to use the gear you have to it's maximum capabilities instead of spending any money. This includes some combination of post processing, positioning, and persistence along with living with the limitations of your gear when you butt up against them (which you'll still do even if you throw $5-10k worth of gear at the issue).
In between, it might be possible to throw some money at the problem and get some of what you want with older long telephoto manual focus glass such as a Super Tachumar 400mm f5.6 and an M42 adaptor for the camera body. You'd loose image stabilization and need a tripod or more likely a monopod. People tend to use one with big image stabilized glass anyway. You'd have to work hard for the shots you want, but it would be possible to shoot end to end on a hockey rink. Total cost with a modestly reasonable monopod and head, you'd probably be around the cost of your camera and lens.
Anyway, it's always going to come down to getting the pictures you can get and compromising on the one's that you can almost get and living with not getting everything. Some subject matter will be well within the technical envelope of your equipment and other subjects will require learning the right creative compromises no matter what equipment you have.
It's worth looking at good hockey photos to get an idea of how photographers handle the limitations of their equipment. Fortunately, the NHL has plenty of images to study.
Few of them are shot from further than across the width of the rink and many of the most dramatic shots are close ups.
Some, like those from aerial and in goal cameras might be possible with non-DSLR cameras such as the GoPro or a cheap remote controlled cell phone and a fair amount of planning and technical infrastructure but relatively little money.
Like most sports, many of the standard subjects don't require particularly fast shutter speeds: faceoffs, goal celebrations ('jube'), impact with the boards, kissing the trophy, line changes, etc.
Relatively few of the action shots stop the puck in mid flight after a slapshot. A fair number are timing shots where the player is changing speed and direction and not moving as quickly across the frame.
Almost no shots are end to end of the far net.
Wide angle lenses provide some of the most dramatic images.
What makes a photograph tell a good story is often written on the player's face rather than evidence that a goal was scored.
As already stated in the comments:
Generally for more close up shots you would need a bigger focal lenght, so a lens with more than 250mm, or a teleconverter.
The teleconverter might be the cheaper of the two, but (as @inkista correctly points out) since hockey is a fast thing to shot, quick autofocus is requiered. And some teleconverters might not (fully) support autofocussing.
You get what you pay for and great camera equipment is not cheap. Canon does not have a better low budget lens than the EF-S 55-250mm with IS.
Without a budget to work with, EF-S 55-250mm image stabilizer is a good cheap lens. "Wow" type images you are accustomed to seeing in hockey magazines and online are taken with super fast, professional telephoto lenses, typically big, white and expensive.
I would recommend a 100-300mm if you didn't have the EF-S 55-250 with image stabilizer, although there is not much advantage with an extra 50mm of focal length.
The next step up is a big jump in price, but you get what you pay for with a Canon EF 100-400mm f/4.5-5.6L IS USM Lens. New this lens retails for $1299 US.
Unfortunately, sports is one of those areas where you may have no choice but to spend a significant amount to get suitable gear. (The other one is wildlife). And if we're talking indoors or night time, then the cost goes up substantially more.
Reach costs. Larger maximum apertures cost. In combination, they cost astronomically more. And right now, you have a "sweet spot" lens that many would consider to be cheap, because lens money is not like other money, alas. In the new-lens world, $300 is cheap, $600 is moderate, and stuff starts getting "expensive" around $1000.
Ideally, most folks would probably state that what you want, really, is a 70-200 f/2.8L lens of some kind (there are a variety on the used market with and without IS, MkI vs. MkII [much improved optics]), which has less reach than your 55-250, but more maximum aperture, which would allow you to get faster shutter speeds to freeze action, and probably pro-quality optics, which are a large step above those of the 55-250, and fast AF capability--especially if paired with a more sophisticated AF system (e.g., an XXD or 7D body).
If you're shooting from the stands, however, then you're more liable to need about 300mm reach, and most 300mm lenses you can handhold from Canon and that are $1500 or less are still going to have a max. aperture of around f/5.6. The 300/4L is US$1350, and the 300/2.8L II is US$6100. The 400/4L DO IS II is $6900. The 200-400/4L IS II is around $11,000. The leaps upwards in price for speed+reach are very steep.
If you can't afford to spend on better gear, then the most you can really work on is getting better access. If you can shoot from rinkside with the ability to move around the rink, that will help enormously. Possibly consider using a monopod. If you can set up some kind of lighting that might help, too.
Currently, I'd also say that possibly considering upgrading your camera body to a used XXD might be worth it from a handling perspective, as well as an AF-speed one. You might want to consider swapping to a USM 70-300 lens, but optically, the performance is likely to be on a par with your 55-250 if not a little worse (the Mk II version of the 70-300 IS USM is relatively unknown at this time, but it's liable to be on-par with the 55-250, the Mk I was a little worse, being much older), and the reach isn't liable to feel like much of an advantage, but AF speed might be better.
You could also consider an old used 70-200 f/4L (non-IS, Mk I), but then you don't get a lot of aperture advantage, and you lose some reach for the L optics and AF performance.