You can reuse old potting soil, many people do, even if its left outside in pots all winter, but there are a couple of points to consider. There may be pathogens present in the used potting soil that you can't tell are there, so there is a relatively small risk that your new bedding plants next year might succumb to something. Given you've got more than one pot full of soil to be stored, I wouldn't empty them out and mix the soils together - just leave the soil in the pots, because that way, you're not combining the soils (and possible different pathogens) from different pots. If you grew anything in any of them this year that developed a problem (root aphids, vine weevil, anything where larvae could be in the potting soil), or if any had fungal infections, dispose of the soil from those pots, don't reuse it.
The next point to consider is this - unless your compost is produced using a hot aerobic method, you should not add it to pots. Cold. anaerobically produced compost is suitable for use on open ground and a valuable addition to the soil, but may contain weed seeds and pathogens which could be harmful (or a nuisance from the seed point of view) contained within a pot or tub. Just use a basic fertilizer, a granular or powder formulation,something with an NPK of around 7-7-7, to 'liven' the potting soil. If you haven't emptied the pots out, but leave the soil in them over winter, then give them a good dig around, turning all the soil within over, extracting roots and checking for bugs,then work the fertilizer in when you want to plant next year.