You should be able to restore items that picked up color with a bleach soaking solution as long as an item is safely bleachable.  First, check the care labels - Don’t use the soaking solution on items that contain wool, silk, mohair, spandex, and leather - These should never be bleached.  Next, you didn’t say if the items that picked up color are white or another color (or combination of colors, so you may need to check for colorfastness to bleach with this simple bleachability test:  dilute 2 teaspoons Clorox® Regular-Bleach in ¼ cup water; apply a drop of this solution to a hidden part of the each item (like a hem, cuff, collar, or inside seam); wait 1 minute then blot dry; no color change means the item can be safely bleached.  Now for the bleach soak:  fully submerge the item in a solution of ¼ cup Clorox® Regular-Bleach diluted in 1 gallon cool water for up to 5 minutes, then rinse thoroughly.  Air dry the item, and check for success (it’s important to keep the item out of the dryer so the heat doesn’t set any remaining dye).  Hopefully the fugitive color will be gone, but if it is lighter, then repeat the bleach soak again.  However, if the bleach soak leaves the dye transfer color unchanged, you may need to try RIT Color Remover, which can usually be found at drug and hardware stores.  A dilute RIT solution works best to limit the stripping-off of the original color of an item in addition to the fugitive dye, so be careful.  Good luck, and let me know how it turns out.