How can I progress further when there are no stairs down?

Nethack has secret passages (blocking corridors) and secret doors (embedded in room walls). You must find them before you can pass through them. The canonical way of discovery is searching, but it's not the only one. For example, if you're sure there's an hidden door in a certain square, you can try kicking it. If you're lucky you'll find the door, break it and gain some bonus experience points -- IIRC.

That's all good and fun, but where should you search?

Stairs always happen in rooms. Thus, you should look for "holes" in the map where a room could fit. If there are dead end corridors in that general direction you might be on the right path!

Here's a real life scenario. I marked some educated guesses with question marks to illustrate my point.

                  -----       -------    ???????
                  |...|       |.....|    ?     ?
                  |....###### |......#   ?  3  ?
                  |{..|      #......|#   ?     ?
  -----------     |...-#######-------#   ???????
  |..........#####|...|              #      ?
  |.........|    #-...|              #      ?
  |`......<.-#####-----              #      ?
  -----------    ##############      #      ?
       ?                      ?      ####   ?
    ?????????              ???????   #--|---?
    ?       ?              ?     ?   @|....|?
    ?   2   ?              ?  1  ?   #|....|?
    ?       ?              ?     ?   #.....|
    ?????????              ???????    |....|
                                      ------

1 looks the most likely: a continuation of a dead end corridor. 2 is also plausible, that large area down there is so likely to have at least two rooms. 3 looks more far fetched, but it's exactly where the hidden door was.

I hope this still gives you an idea of how to make reasonable guesses when looking for secret passages.

Oh, and remember to never ever hold a key when playing Nethack. One search is hardly going to be enough, so run the search command multiple times the right, safe way: 20s (020s if you have numpad on) will search the current position for 20 times only if it is safe enough to do so. Hunger, approaching monsters and the like will abort your search, unlike holding a key does.

You're right, Nethack contains hidden doors. There are sometimes the only way to progress further in the game.

The default key binding for this is 's'. Take a look at the "Tips" section of this small introduction: Nethack - First Steps

One of the best things about nethack: there are almost always more solutions! It's true that once in a while you will simply die before you find the stairs. But that particular end should be very rare, and if you find it common then definitely look harder for other things you can do.

• Monsters do keep spawning. If they seem to stop coming for too long on your current level, go up one. Keep killing monsters, eventually enough items may drop that you can use some of them to make progress.

• Scroll of teleport or wand of teleport. If you don't have a scroll but you do have a magic marker...

• wand of digging or pickaxe or dwarven mattock: sure, it's fine to go digging horizontally and find the stairs, but you can also just dig down ('>' for the direction key).

• it's work snooping around even in rooms that don't look (map-wise) anywhere likely to be near undiscovered rooms. You may stumble into a hole, a teleport trap, or a level-teleport trap, suddenly useful. Or (not so great, but can work) read a cursed or too-high-level spellbook and potentially suffer teleportation as a consequence. Hopefully then you're in a room with stairs!

• polymorph yourself into a monster that can phase through stone, if you don't have a pickaxe.

If you simply cannot get below the current level after trying everything, just continue killing monsters here or on levels above. You shouldn't die of hunger as long as you understand the mechanics of prayer. Good luck!