You don't need the Smash Strike to get out of the Abandoned Cavern, although you DO need it to reach the Explorer who's in there. Thus, when you come back later to retrieve the Shaman and the Idol in the Crystal Cavern (both of those require TNT) you can rescue the Explorer as well.
To get out of the Abandoned Cavern without the Smash Strike,Spoiler Alert!go back to where the big rock slams up and down (the way out to Crystal Cavern). Get up on top of that rock by using the same rope you used to reach the Crystal Cavern. Then turn back around to face towards the rope you just used. To your left (roughly the 10:00 position on a clock) you'll see another rope. Jump to it when the big rock is at the top of its up-down cycle. Once you're on the rope, look up and slightly to your left. There are some twisted metal platforms with a broken ladder on the right side. You need to jump up there; aim for the darker area to the left, and be prepared that Harry can't land on the platform but can reach it and pull himself up.
Once you're on the platform, look to your left. There are more platforms, some with boxes on them, that you can work your way across. Soon you'll find an Idol, which confirms that you're going the right way. Once you get the Idol, look down from the far side of that platform. You'll see a flat area with a porcupine; this is the area with the swinging boxes that you passed before on your way through here the first time. You can jump on down there. From there, you need to find your way to the exit.....but I'll let you try that on your own before posting how to do it. Come back if you need more help. (BTW, the Abandoned Cavern contains two Idols and one Explorer. The Underground Dam contains two Idols and a Shaman. So if you have 10 Idols, or will have 10 Idols if you grab all these, and if you didn't visit the Shaman in the Underground Dam on the way in, you could buy the Smash Strike from him after you exit the Abandoned Cavern, then come back and get the Explorer. Or you could get the Explorer later once you have TNT.)
BTW, I assume you got the Pickaxes already (you didn't mention them). If you didn't, then you need to go back the way you came. You're going back into the mountains and you won't be able to get anywhere without the Pickaxes.