Much in the way a predicate about predicates is a second-order predicate, a second-order disposition is a disposition involving dispositions. This is an instance of what might be called 'conceptual self-reference' or 'recursive thinking'.
Let's make sure we have a good understanding of the term 'disposition' to help us disambiguate. A good place to start is SEP: Dispositions. From there:
A glass has certain dispositions, for example the disposition to shatter when struck. But what is this disposition? It seems on the one hand to be a perfectly real property, a genuine respect of similarity common to glasses, china cups, and anything else fragile. Yet on the other hand, the glass’s disposition seems mysterious, ‘ethereal’ (as Goodman (1954) put it) in a way that, say, its size and shape are not. For its disposition, it seems, has to do only with its possibly shattering in certain conditions. In general, it seems that nothing about the actual behaviour of an object is ever necessary for it to have the dispositions it has. Many objects differ from one another with respect to their dispositions in virtue of their merely possible behaviours, and this is a mysterious way for objects to differ.
So, in less technical language, a disposition is a tendency of something to do something generally under general conditions. This of course is very broad! Concretely, the disposition of a glass (the something) to shatter (to do something) when dropped (general condition) is a disposition.
Objects are dispositionally simple because they tend to have more reliable outcomes in the same general conditions. Think about a trial of dropping a glass 1,000 times. How likely is it to shatter? It's not deterministically forgone because if the physics aren't right or there are exceptions to the general condition, the glass won't break. For instance, perhaps a pillow falls from the couch at the moment the glass is about to strike tile and it is spared, or perhaps the borosilicate composition is such that it is shatter-resistant.
Most common objects are reliably dispositional. Dropping fragile objects onto hard surfaces leads to the same ends. But there is a class of entities that are much less amenable to such dispositional analysis: agents (SEP: Agency).
In very general terms, an agent is a being with the capacity to act, and ‘agency’ denotes the exercise or manifestation of this capacity. The philosophy of action provides us with a standard conception and a standard theory of action. The former construes action in terms of intentionality, the latter explains the intentionality of action in terms of causation by the agent’s mental states and events.
Simply put, an agent possesses a degree of SEP: intentionality which is generally understood that it can be "about" things around it, that it is aware of dispositions including intentionality of other agents. Thus, when one agent has dispositional awareness of another agent's dispositions, we have the second-order disposition.
Let's examine a game like poker where agents compete against each other with randomized cards with game-theoretic means to achieve the goal of winning. While your opponent may know the rules and even the probabilistic frequency of the cards, one element involved in poker is the ability to bluff. This where is meta-dispositional thinking comes in handy.
For example, you know your opponent isn't good at lying and has "tells". For weeks, you may be able to "read" your opponent, and you repeatedly win the game. But after reading a book on tells and lying, she suddenly understands privately the mistakes she has been making. Now, she might, in her bid to win (both), understand that you have the disposition of betting as if she isn't good at lying and has tells. She may now cleverly PRETEND she is no longer good at lying and use her tells consciously to win. This is her new disposition about your dispositions. A second-order disposition (or third-order if her disposition is about your disposition about her dispositions, and so on.)
Of course, the understanding of meta-dispositional thinking is critically important from a game-theoretic perspective. National security is based on the use of intelligence and counterintelligence. Allowing one of your intelligence assets in the field who is compromised to continue operating as a backchannel to the enemy may be more useful as a channel for introducing disinformation directly to their handler. Note, that this relies on a second-order disposition. The handler may be working from and acting on the dispositions of the compromised asset. Thus a handler's dispositions are derived from the dispositions of the double-agent, and your dispositions are second-order relative to the presumed dispositions of the double-agent's handler.