In your original post you said you guessed your stabilizer is low because you normally start the season with low CYA. Let's therefore say it's under 30 ppm, assuming you haven't used trichlor pucks or added any stabilizer compound.

Any chlorine you put in the water will be subject to degradation by UV light in the absence of stabilizer. It still works to kill algae but you'll need more chlorine because much of it is being consumed by sunlight before it can do its job.

If it were my pool I would take a sample to the pool store and have them test for CYA. I would buy a small bucket of stabilizer and add one half the amount needed to get to the CYA level I like to maintain. Use the Pool Calculator at http://www.poolcalculator.com/ to figure out how much you need. I would also get some R-0013 or, as I said earlier, a CYA test so I can keep tracking those numbers. Leslie's sells a product called Blue Devil, a CYA testing kit.

You can clear up your pool without stabilizer but it'll take more bleach and more time without stabilizer in the water.

ETA: I had tried to post this earlier but the forum was down. Watermom's approach and mine differ slightly; choose whichever feels right to you. In any case, make an effort to get a refill for your CYA test. I bought a pint from www.amantoind.com which will last me a couple of seasons. Unless you have a pool store nearby and trust their testing you cannot get by without knowing your CYA level.