Jams aren't just broken lead. They often include a chunk of lead wedged in the jaws of the mechanism, and may include lead debris and powder that have been compacted in the feed tube or mechanism. They don't easily come out just by pushing new lead behind them. If you are able to push the plug out but don't clean out all of the debris, you're already on the way to the next jam.
For completeness, I'll try to cover all the options, because even with a cleaning tool, some methods are more effective than others. So only portions of this answer will address your current predicament.
Cleaning tool
The 0.3 mm size is pretty small, so typical household items that work for larger pencil sizes may not fit. It happens that 0.3 mm stainless steel wire is commonly used for various outdoor purposes and can be found at many sources. It is easily bent, but it's a lot stiffer than the equivalent copper wire, and short lengths will stay straight as you push them through the pencil if you're careful.
The question says the pencil has "been clogged for a while now". If you don't have a suitable cleaning tool at home, let it be a while longer until you can find one. But if you want to exhaust your options for using it now, the Disassembly section, below, might help.
The normal way to clear a jam is by plunging it out with a cleaning tool. And you can do that with the mechanism fully assembled or disassembled; there are some implications. But if you can disassemble it, you can sometimes clear it without a cleaning tool.
Disassembly
Disassembling the mechanism is a better way to thoroughly clean jams. You can see what you're doing, you have access to everything that needs cleaning, and you can get all the debris completely out of the pencil. Even if you don't have the right cleaning tool, you can often make due with other things when the mechanism is apart.
Sometimes once it's apart, you can see that the jam is caused by a piece of lead wedged in the jaws, and you can dislodge it without a cleaning tool. Or you can clear enough of the jam to use something like another lead to gently push out what's left (keep a few of the hardest leads you can find for this purpose if you can't find the right cleaning tool).
From the picture, it looks like the mechanism isn't one that comes apart just by gravity or unscrewing something obvious. But they typically aren't assembled in a non-reversible way, like by parts snapping together permanently, or gluing or welding the mechanism together. Usually, something unscrews or they're just a tight friction fit. If you don't have disassembly instructions, a few things to try:
- On many pencils, the metal cone containing the nozzle unscrews to reveal the mechanism. Try unscrewing that, using something like a flat rubber band for friction if you can't grab it well.
- If the cone doesn't unscrew, try grabbing the lead loading tube and turning it to see if it unscrews the mechanism from the tip (this would be an unusual method of assembly, but it can't hurt to try, just don't force it).
- Try pulling the lead loading tube out of the tip with some gentle force (the strength in your fingers). Rather than just yanking in opposite directions using the strength in your arm muscles, grab the parts and push them apart from each other in a controlled way with your fingers. Do it on a big, white surface so if it suddenly pulls apart, no important pieces will disappear. Recognize that no friction fit should take unreasonable force to pull apart. If reasonable finger force doesn't do it, something probably unscrews.
Cleaning while fully assembled
People usually do this by pushing the cleaning tool into the nozzle to push the plug out the back of the mechanism. It's fast, and often gets you going again in a hurry, but it isn't a good idea. That just temporarily moves the debris to a place where it will get pushed back in, and may leave debris stuck in the mechanism.
If it's your only option, it's better to use the correct size tool--some kind of sturdy pin or wire that snugly fits in the nozzle. Something that doesn't fit snugly may push out the clog, but leave debris behind, even compacting it in nooks and crannies.
First empty out all the lead behind the mechanism. If you manage to push the debris out the back, get it out of the lead loading tube using something like a pipe cleaner, or if you can remove the tube, you may be able to blow it out.
If you can't disassemble it, the better way to do it would be with a long cleaning tool inserted through the lead loading tube. Something that long will easily bend when you push on it, so you need either something very stiff, or make a tool by enclosing all but the end of the wire in a tight-fitting tube that fits inside the lead loading tube.
Another approach is securing a cleaning pin in the tip of a tube or wooden stick that fits in the lead loading tube; something like a WD40 spray tube or a thin wooden stirring stick; you may need to hunt for something that fits.
If an undersized wire is all you have, you won't be able to clear out all the debris well. But if you're using a snug-fitting cleaning tool from the back end, you can get some of the debris out of the jaws. After pushing out the plug from behind, push the cleaning tool in far enough to spread the jaws, tap the mechanism to jar the debris from between them, then push the tool through to plunge the debris out. Repeat that a few times.
Consider an alternative pencil
All that said, I think you're in a losing battle using that pencil for 0.3 mm writing. I've tried 0.3 mm pencils several times in the past because of the thin line they produce. But the leads are delicate and notorious for breaking, and softer leads that can make a dark line tend to crumble inside and cause jams. If you want to use 0.3 mm, it's only practical if you use a pencil with a mechanism you can easily disassemble for the frequently needed unplugging and maintenance.
Within the last few year, some Japanese manufacturers have come out with some clever mechanisms that minimize breakage. I haven't tried them, myself, but various articles and reviews I've read report that they are better at avoiding or reducing breakage and jams. A video by retailer JetPens, Why You NEED an Overengineered Japanese Mechanical Pencil!, describes some of the mechanisms, although I didn't check to see if every one is available in 0.3 mm.
One mechanism takes a different approach. Rather than trying to avoid breakage of a 0.3 mm lead, it uses a larger, stronger lead, but rotates it to keep a sharp point on it.
If I get the urge to try 0.3 mm again, I would probably only consider a pencil with a mechanism that makes it practical.
