It sounds like your flowerbed has already been amended (at one time) if it is already built up to the level of the edging. If your bed has a layer of mulch that is applied annually, that mulch breaks down to feed the soil. You can add some organic fertilizer to help your plants since there isn't room to add a lot of compost. It would be a huge job to shovel out a bunch of the soil to mix in that much compost since you would need to dig up plants, then turn around and replace them. Adding the 6 inches of compost is for making new beds. Yours is an existing bed, so keep that layer of mulch on top and add some organic fertilizer.

I will often just sprinkle some compost or manure around my plants on top of the mulch. You can pull it back to do this if you wish, I'm just too lazy. Do you have mulch?

Other things to consider are, compost tea, fish emulsion or seaweed emulsion spray, fish emulsion granules ? (don't know the proper name for them...)

There are many organic things on the market these days that can help out your soil without adding bulk. Just depends on what the soil needs. Bone meal, blood meal, copperas, etc.

I bet a few people here could give you a long list.