Don't put so much emphasis on precise Initiative
People assume the danger is over because they assume the only danger comes from combat, and combat starts with initiative. If you start every combat with an initiative roll, everyone will know when to expect danger.
But if you use a single initiative roll for all encounters (rerolling initiative at the end of a long/short rest), you can jump into combat on a whim without the players getting too comfortable knowing when the danger/combat is over.
Swinginess
There are going to be a few moments where a Rogue is frustrated with an initiative of 6, but you can easily justify those moments with some good storytelling.
Example:
"Bob the Rogue (initiative 6), something about this place is extremely
unnerving to you and your constant state of alertness is working
against you, making you jumpy and distracted. However, Johnny the
Paladin (initiative 16) is ready to bring the light to the darkness,
and is the first to act in this den of evil after his short rest."
The number is set, so it's pretty easy to ask yourself why a bard might have a low initiative (hangover), or you could ask your players to fill in why their initiatives are so low.
You can provide one-time initiative bonuses/penalties to mix it up ("since you're still dazed from that Sleep trap, Bob, your initiative is -5 until your next rest"). I have allowed players to use Inspiration to reroll their initiative, and keep the new roll if it is higher, although they'll usually just prefer to use it on something else.
Players with lower initiatives will feel like they don't get nearly as many turns, so make sure to mix it up with minor adjustments fairly frequently.
Bonuses
I have found that most players will prepare during their rest to improve their initiative rolls (since you bought that map from that shady merchant, the party knows the layout of the area and is more focused on individual details. Everyone's initiative gets a +3). This also puts a lot more emphasis on sleeping conditions, rather than relying on Exhaustion rules. This seems like a big deal, but it mostly just gives off the idea that their minor actions matter.
I did not have an Assassin Rogue, so I did not have any issues in that regard, but IMO the Assassin Rogue is difficult enough to play, and another option for players to feel penalties/bonuses that they deserve is a welcome addition. If an assassin buys a map and gets his sleep and eats his Wheaties before his big day, why shouldn't he deserve a +6 initiative for his contract?
Penalties
On the opposite side of the spectrum, penalties can (and should) be harsh. "That big fight you guys had last night took its toll on you. You finish your long rest, since you all went back to sleep, but it was restless at best. I want you all to roll your initiatives for this rest with disadvantage".
Tracking
Initiative isn't hard to track, and since it doesn't come up too often, modifying it here and there doesn't make it nearly as difficult as tracking health of baddies. I have found this does require a little more work, with players forgetting to apply the penalties/bonuses or with myself having to track a little more information, but I find it to be worth it, since the alternative is static and uninteresting regular initiative rolls.
After Combat
Once the "obvious" combat is over, say so, and attempt to 'round-robin' around the players to get their actions in a rough initiative order, almost like in combat. You can continue to roughly track the "rounds" if you need to, but the goal here is to transition smoothly between in-and-out of combat. Don't bother asking for speed or distance or anything specific, just assume that they're moving their maximum speed. Not everyone needs to make an action (since they'll just be following the leader like normal), but make sure to ask them the first few rounds after combat anyway. They are choosing to do nothing for their time, rather than them just not existing during RP moments.
Example
"Johnny, with that blow, the shadow fades out of existence, and the room has a brief moment of tense silence. Do you take further action?"
"I walk over to the statue it was guarding to investigate it" (Statue is 70 feet away)
"Johnny is walking towards the gargoyle statue, which in the darkness almost seems like it is watching him. Bob, do you have something?" (Bob being next in initiative)
"I ready my bow to shoot"
"Noted. Martha? Walking next to Johnny, alright. James? Looting the bodies... got it. Johnny, as you approach the statue..."
Treat it not that combat just ended, but that the last noticeable target was just killed. You can continue to keep a combat-esc pace without it feeling clunky, as long as you aren't too strict on timing/turns/etc. The more "rounds" of this you go, the less the initiative order should matter, but keep at it at least for the first couple rounds after.
If things happen out of turn (James loots a body that's not dead! But Johnny's first in initiative) you just ask what the character with the highest initiative what his response is and take it from the top.
Your players will try some shenanigans with holding their Actions while "out of combat" (mine definitely did), but I simply just treat that as them having an initiative of +10 for the first round of the next "combat scenario" if there's no "active combat" going on. Otherwise, if you let them use held actions as free attacks before combat, they're almost getting a free surprise round every encounter, and surprise is something to be earned, not cheated.
Notes
Maintaining the feel that their actions and time matter, by utilizing a persistent initiative, will create a sense of tension that normally ends because combat feels like an on-off switch. Rather than forcing tension that players usually feel only in combat, make it feel like danger (combat) is always just around the corner.
My players found it to be a very positive change. They were a little confused at first as to knowing when combat ended/started, and who's turn it was out of combat, but this was resolved once they got into the flow of things.
