Heya,
I agree with the commentary about changing how you approach getting to your subjects and photographing them. We all inevitably want longer, faster glass, and better cameras for going into tight, dark places and shooting something small and fleeting. Even with a $20,000 super tele setup with the latest tech, you'll be challenged to do this. You could keep throwing money at the problem as a solution but it will not result in what you think it will directly. What will change your results right now, however, is how you approach getting the shot in the first place.
You say you don't use a hide or anything, but why not? You can get easy dog-house style pop-up blinds for $50 that fold down into a disc basically. I use one of these as my mobile hide and I get within 8~9 feet of song birds with my 600mm. In really low light, hand holding is going to cost you a fortune, while a simple set of legs & gimbal will free up your shutter speed and ISO ceiling quite a bit. I get as low as 1/50s on a good stable setup, at 600mm on APS-C and APS-H, while still using ISO's as high as 800~1600, it's that dark! So I can appreciate what you're trying to shoot in. A mobile hide and a light weight mount setup will give you so much more in wildlife photography than simply getting faster but shorter glass, or a faster, lower resolution, lower ISO performing camera (1D3).
Glass wise, you're good where you are. The only way to make a big difference on that Canon mount, is to get a significantly larger aperture. F5.6 and F6.3 are next to no different. Not even worth comparing. F4 would be the only starting point frankly. This is where a 500 F4L (mkI, early version, no IS) comes into play around $2k. You don't need IS as much if you're using a mount--which you should at this point probably. Other options, include a 300 F2.8L (non IS, MKI) around $1800~2k. You put a 1.4x TC on there, and you have a very good super tele. And you have a great BIF lens while you're at it. Or, bare at 300 F2.8 on APS-C is great too. Use that 20MP resolution to your advantage and get into some really low light with F2.8 and drag the shutter using a tripod. And finally there is the Sigma 120-300 F2.8 OS (non-SPORT) around $2k as well. Takes TC's as well. These solutions all get you F2.8 or F4, or both, and can get up to 600mm. These would be the glass solutions in that budget range that I'd consider. Moving from a Tamron 150/600 to a Sigma 150/600 will change virtually nothing for how and what you're shooting.
But even with those as options, I really don't think it will change things as drastically as simply changing how you approach your subjects.
A mobile hide & a tripod can fit to a backpack, and weigh less than a super tele lens does.
Study your subjects.
Go to that location before they're active.
Setup shop. Get your shots.
The Tamron 150-600 has lovely bokeh. I can get totally blown out backgrounds with buttery bokeh, at F7.1 or F8 no problem with the same exact lens you're using, take a look below. I'm using significantly inferior cameras compared to a 70D too. The difference is, I get closer and I use a tripod in pretty dark low light. The other difference, is I over-expose my subjects in this kind of light by over 1 and 1/3rd stops and bring it down in post. It makes a big difference. That's all! Tom guided me through this process, so I can show you my results just from listening to someone who does this for a living.
Here's the blind I use (link)
. It folds into a disc, just like photography lighting reflectors do. It has no bottom. Just the 4 walls and each side can be opened to look out and point a lens. I sit inside with a little travel stool, like this one (link)
.
For laying-on-the-ground approaches, a drape works instead (like at fresh water holes, shoreline, etc).
Lastly, if the light is lower than what I shoot in, you can add flash for fill.
Here's an ancient 1D2 at ISO 640, shutter 1/320s at 600mm and F7.1 from the same lens you're using.

IMAGE LINK: https://flic.kr/p/rg8BAb
LE1M1615
by
Martin Wise
, on Flickr
Here's a humble Rebel at ISO 800, shutter 1/160s at approx 300mm and F7.1 from the same lens you're using.
IMAGE LINK: https://flic.kr/p/rqFnEt
IMG_3849
by
Martin Wise
, on Flickr
Just get closer and use a hide & mount. You can get amazing bokeh and great light in low light no problem with lesser setups with that same lens!
Very best,