The only trick involving the ending of the numbers are ones like the following:
$$\begin{align} \gcd(1234567, 3210987) &= \gcd(1234567, 3210987 - 1234567)
\\&= \gcd(1234567, 1976420)
\\&= \gcd(1234567, 197642)
\\&= \gcd(1234567, 98821)
\end{align}$$
The last two steps take advantage of the fact the first term isn't divisible by 10 or by 2, so I could cancel them out of the second term. Computer implementations often use this trick to pull off the factors of 2 after each step. (Factors of 10 are not useful to the computer, and for a human to make use of it, he either has to get lucky or do extra work to force a 0 to appear)
Examples of forcing a 0 in would be something like
$$\begin{align}
\gcd(1234567, 98821) &= \gcd(1234567 - 7 \cdot 98821, 98821)
\\ &= \gcd(542820, 98821)
\\ &= \gcd(54282, 98821)
\\ &= \gcd(27141, 98821)
\\ &= \gcd(27141, 71680)
\\ &= \gcd(27141, 7168)
\\ &= \gcd(4 \cdot 27141 - 3 \cdot 7168, 1 \cdot 27141 - 1 \cdot 7168) \qquad \qquad (*)
\\ &= \gcd(87060, 19973)
\\ &= \gcd(8706, 19973)
\end{align}$$
The step (*) needs elaboration: several steps of adding/subtracting one from the other were combined into a single matrix operation
$$
\left( \begin{matrix} 4 & -3
\\ 1 & -1 \end{matrix} \right)
\left(\begin{matrix} 27141
\\ 7168 \end{matrix} \right)
$$
The main thing is that the determinant is $\pm 1$, so that it really is just a combination of adding / subtracting one row from the other and swapping them. I picked $(4,-3)$ just to have smallish numbers that produced a $0$ at the end. One could try something more systematic....
This trick is also used in computer implementations for sufficiently large numbers, but usually they trick is applied to the left end rather than the right end (Lehmer's algorithm). And as is often the case, ways to make things easier on the computer don't always translate into ways to make things easier on humans, so I can't guarantee you'll actually find this useful.
There are other tricks like splitting the two numbers into left and right halves (the "half-GCD" algorithm), but you're trading large GCD's for large multiplies, and will almost surely not be useful for a human.
Really, the only "tricks" of the sort I think you have in mind are eyeballing special forms that won't work for general computation. e.g. to compute
$$ \gcd(100000100101, 1001) $$
You might left shift 1001 up to cancel off the first 1, then add in another shifted copy to cancel the -1:
$$\begin{array}{r}
100000100101
\\ - 100100000000
\\ + 100100000
\\\hline
\\ = 200101\end{array}$$
and so
$$ \begin{align}\gcd(100000100101, 1001) &= \gcd(200101, 1001)
\\ &= \gcd(100001, 1001) \end{align}
$$
And do it again (but also left shift 100001 so we have room:
$$\begin{array}{r}
\\ 1000010
\\ - 1001000
\\ + 1001
\\\hline
\\ = 11\end{array}$$
and so to continue the computation:
$$ \cdots = \gcd(11, 1001) = \gcd(11, 11) = 11$$
Or other cases where you can quickly spot something you can do to let you multiply or divide things very quickly, or cancel things out to simply stuff.