If one flight departs 4 hours after the other, it appears that your destination is served with a far greater frequency; as such, that the first possible connection would require a minimum waiting time of 11 hours seems highly unlikely.
If there are earlier flights, regardless of whether they are far more expensive, or even indeed sold out, then you won't be entitled to a hotel by lengthening your stay to obtain the cheapest possible prices. (The inconvenience is the reason they are charged like this; this doesn't automatically mean you get a free hotel. The free hotel is offered to make up for the inconvenience of having no alternative to the long connection; it is not there to make your choice of a far-cheaper connection more personally palatable)