How to play a microphone out of a speaker

Don't be so shy.

0.3W is pretty loud! And don't forget that audio equipment potentiometers are logarithmical ones, so when you turn it halfway, it is roughly one tenth of the output power. So a 20W amp, driven with its rated RMS input signal would output 2W RMS when turned up halfway. A quarter setting would result in 0.2W RMS accordingly. And you said you'd want to test with music - rated music power is around 2x the rated RMS power for most of the speakers.

Don't fret, speakers are quite difficult to destroy in so low power situations. I did try with a PC speaker and a ~20W amp when I was young and feeble, to the point when it got quite warm on the outside - but no pops (it was horrible, by the way)... Ok, PC speakers are not too flimsy. I did build a tube amp from scrap parts, and on that I regularly used a way too -electrically- underrated speaker, and to make the situation even worse, I used it as a guitar amplifier - driven into the maximal power tube saturation I could achieve...

If however you'd want a professional approach, use an LM386 power op-amp to play with, a few external components and you are ready to go. I think that is far easier to obtain than an impedance matcher transformator - not to mention, it degrades and colours the sound a lot less.

LM386 datasheet

Your problem may be that the LM358 is not designed to drive a speaker as a load. You will need to add an output stage with a low output impedance of ~4 to 32 ohms or less, depending on your speaker's impedance. Here is an example of a typical configuration using BJTs as the output stage:

Note how the feedback loop has been modified to incorporate the output stage. A trimpot is used in this schematic to adjust the volume.

My second hypothesis, is in the event that you have a high impedance set of headphones (around 600 ohms), is that your gain is too high and your are railing out. If that's the case, then you can either reduce your gain or replace your gain resistors with a trimpot and try adjusting the volume until it sounds better. If the "rett rett" sound you described is a loud sound, then I would recommend skipping my first suggestion and try this first instead.