Currently its maximum is at 6 cards (I believe it gives a new card every 10 minutes, so that's an hour worth of cards). Is there any way to increase this number to something higher?
No, there is no way to increase your maximum opportunity deck size. Even if you become an Exceptional Friend by paying 20 Nex a month, your deck size will not increase.
This means that if you want to draw as many cards as possible, you have to return to Fallen London once an hour.
This has changed this month; being an Exceptional Friend now grants you a maximum of 10 cards in the deck (up from 6).
Other than that, it's still impossible to change it.
There is an exception for some special Locations where your deck is unlimited (such as zailing the Zee), but that usually means that cards are non-discardable and you are still bound by the number of actions.