I nerd out on alot of dvds so ill throw out some various details i've picked up along the way, these serve me very well but I take no responsibility for any weaknesses in them:
I was told from when I started jiujitsu that you don't need the arm across if your doing the triangle right, so I never considered it a priority. Sometimes in tough battles when I don't have the angle right i'll resort to it, but I think there are other adjustments that can make a decapitating triangle without it.
First and foremost, the golden rule : Knee Pinches + Leg Curls.
Some simple adjustments i've seen in various DVD's or learned at seminars:
When locking the triangle, try to make the leg that's locking your other leg go OVER their shoulder -- the same rule as with armbars: you want to make their shoulder invisible. If you accomplish this, even with their arm on the outside their shoulder will still be pressuring into their neck.
When you do have a triangle locked, and as a good tactic when/if they start to posture/stack, walk yourself backwards with your elbows -- this brings their head closer to your hipline from chest, reducing their posture, their ability to stack, and making the triangle considerably tighter.
The leg that ISNT around their neck, is generally stronger aligned with their back, not going away from the opponent against the side of their body -- the knee squeeze becomes more hip dependent at this angle, and uses less leg muscles.
When you first attempt to apply the triangle, the leg trapping their neck should come down at angle of their neck, not the angle of their shoulders, this creates a much stronger initial pressure and makes securing the triangle easier. Some of the 10p black belts it feels like your being clubbed in the back of the head when their leg comes down on your neck for the triangle.
The single thing that generally makes triangles win or lose for me, is the angle of your legs/body compared to them. As Ryan Hall puts it in his Triangle DVD's "its called a tri-angle, not a tri-lay in front of the guy"(even though triangle is really referring to your legs, still humerous

). The principal is that you want the bottom of your leg, the underside/hamstring to be flat on their neck, not the inside of your thigh, the difference this creates is massive - both in tightness, and in the power you can generate in the squeeze. The ideal way to accomplish this is to cut your body perpendicular to the opponent in the direction of the side the leg around their neck is. So if your right leg is the one around their neck, you cut to the right side. This motion is difficult at first to execute while holding the triangle tight, but comes with time. Something that helps alot AND stops a common counter from here (the counter is them stepping over your head) is to grab their arm, orif you can, leg, on the side your cutting too, this keeps them in tighter in the triangle and stops them from having the ability to stack you, or to try the step over escape.
My favorite and highest percentage submission, and
possibly the tightest form of triangle choke in existence is the
Omaha Twist an adjustment made by Ant from Omaha. It turns it into a choke tight enough to pop a head off with strong neck crank elements to boot. It takes long limbs, or a small core'd opponent to execute, but once you've cut the angle instead of hooking their arm or leg, your reach under their torso with the hooking arm (kunckles facing their body) and sgrip around their torso and squeeze. Both in the gym and in comp, I've never seen anyone escape or last more than a few seconds once the Twist is locked.
If they stack you after you've cut the angle, you can use the 'corkscrew armbar' motion Denny shows in his corkscrew armbar video (except with the triangle, instead of arm) to spin them into a mounted triangle and reverse the stack. Although an easier and quicker option is to walk backwards with your elbows but once your too stacked that's difficult.
An Interesting Option for when their arm is around your hip/back so theres no chance you will get it across (sounds like your description) is you can reverse the triangle so its on the other side / other leg figure four'd. This changes the pressure so their arm works against them I've been shown this by multiple people, but I don't personally go for it as I work the other adjustments I mentioned and don't want to give them that one second of my triangle being released during the transition.
I've got a few more if your interested, but I don't want to make too big a wall of text in one post as I have an often critiqued habit of writing too large of posts...