To reiterate what was already said in the comments: what you're calling "attaching" is usually called "concatenation".
The results of a concatenation depend on the base being used. I suspect that concatenation of primes in binary comes closest to having a satisfying "general rule." In any other base, what you get from concatenating primes is likely to feel like an accident of the base.
In decimal, for example, consider the prime quadruplet 11, 13, 17, 19. Of course 1 was never a prime number, though we didn't realize that until very recently. And 9 is obviously not prime. So there you have four primes in a row that are not the concatenation of smaller primes.
And then we get 23. Nice, huh? The next one is 37, which of course you already found. The next one after that is 53... do you care if the concatenated primes are in ascending order? If you require that, then you're not going to like 73 or 113.
I'm regarding 113 as the concatenation of 11 and 3, but it's also possible to regard it as the concatenation of 11 and 13. I disagree with that. My point is that a lot of situations arise in which subtle differences of opinion lead to different interpretations of concatenations.
Or maybe not so much in this particular instance? I see 137 as the concatenation of 13 and 7, but you might see it as 13 and 37.
This might be enough for a decent OEIS search. I get six results, the first of which is http://oeis.org/A105184 But neither in that result nor the other five results do I find what I would call a "general rule". Do you?