Re-Balance the Long Rest Mechanics
This answer suggests changing the Long Rest mechanics, but without (it seems) too much direct knowledge. I've done exactly that and have extensive experience with it, so I can give some insight into what I did, why I did it, and how it worked. (Although not actually in that order.)
That will give you some insight into how you might want to tinker with your mechanics to get the effect you want (if this is even suitable for you.)
Why I did it:
Basically, I wanted to run a game with somewhat fewer combat encounters and more non-combat encounters and more non-combat role-playing than the 4e rules suggested. This has little to do with any perceived flaw in the combat system (we all liked the positionality of it, even the less combat-oriented players) and everything to do with us: We over-analyze.
At the time, shortly after release, it seemed to me that the default assumption of the game was about four combats per extended rest and somewhere in the neighborhood of eight to ten encounters per level. Indeed, looking at my notes right now I see the sentence, "There is no way I am running a game at that pace." (Unstated: Because we'd never see level two.)
Looking at more recent, but unofficial, sources, I see other estimates at four to six encounters per extended rest, and in hindsight I think five is about the right number, at least for our group.
What I did:
Initially, I changed the Daily Powers (and healing surges and everything else extended-rest-limited) to be Level Powers, recharging at each new level. Simple, clean, easy to remember, no special cases. I then added a lot more extended skill challenges (I love that concept) and a lot more social time, and re-balanced things so that 4-5 combats along with everything else would get them a new level.
I did this-- imposed this, really-- at the beginning of the game.
A little later, I made one modification: Level Powers could be used twice per level, not once per level. (But not all Level-limited effects, just the actual Daily/Level powers meant for combat.) I don't have any notes saying I added more combats to compensate. I think I did not, but possibly should have.
How It Worked:
Fairly well, I think. No one complained, and even though I wasn't very flexible about it, I feel sure my group would have complained had it not worked for them.
The main reason to do this in the first place was not because of resource-hoarding or profligate resource expenditure, but it did have the effect that, when planning an adventure out, I did not have to spend much effort rationalizing things so that combat encounters came in neat bunches per day, or contrive reasons why the PCs could not rest.
This made adventures that involved long-ish overland journeys fairly easy:
Three combat encounters, three hundred miles apart is now just the same as three combat encounters a day.
I can now design smaller outposts of foes, with only enough enemies for a single combat, instead of stringing things together in bigger chunks.
Heck, I could send them on one-shots and let them come back to home base (a city with amenities) without having them fully recharge.
Another way to think of this is that the powers are now paced to the players' advancement and the narrative of the campaign, rather than forcing the narrative of the campaign to be pegged to single calendar days.
One mild downside, however, is that in some sense, the players don't get to "enjoy" or "experience" their newly acquired powers as often as they otherwise would. (In fact, only 40 to 50% as much.) This wasn't a huge deal, but it was the motivation for changing Level Power frequency as I outlined above.
What Can You Do?
You can do exactly what I did, if the reduction in combat is something that also appeals to you. If not, then you probably shouldn't do exactly that, but this at least gives you a candidate solution: Contrive, somehow, to keep their dailies from refreshing until about 5-ish combats have gone by. This might be by GM fiat, this might be by narrative structure ("when the arc is complete," and all arcs always take 4-6 combats) or something similar.
One thing to note, though, is that this is a Very Large Mechanical Change to impose after a game has begun. I would be more than a little put out by a GM imposing that after a game has begun. (I imposed it, but cleanly, at the start of the game.) You're probably going to have to get some player buy-in for this. Which in turn means that perhaps other soft approaches ("Hey, guys, can you... stop doing this? This is not in the spirit of the rules, you know.") might work and should probably be tried first.