New spider veins after laser treatment. Will these go away?

Spider veins are typically hereditary, but it sounds like they appeared after having a varicose vein ablation. Sclerotherapy treatment would remove spider veins, causing them to collapse and fade from view. Assuming it’s more of a cosmetic concern, this is an easy procedure that usually takes around 30 minutes to complete with no down time. It doesn’t require any anesthesia as patients often feel very little pain. 

What you are describing sounds like Telangiectatic Matting which is essentially dilated little vessels due to an inflammatory response from prior treatment.  This is something that can happen and cannot be predicted.  Often, these will go away on their own but it can take a long time (a year or more) and sometimes they won't. You can also treat these with sclerotherapy or laser. 

Lasers for telangiectatic matting or a different sclerosant solution will help clear these veins. They are often very fine, i.e. small in size, < 0.1 mm in size. Remember to wear compression stockings after sclerotherapy (see link below). 

New spider vein formation following sclerotherapy is called neovascularization and can occur following sclerotherapy.  Sometimes it subsides on its own.  I usually treat this by another session of sclerotherapy and often times I will use a different kind of sclerotherapy solution for this such as glycerine which works well for neovascularization.

Sometimes new spider veins can form due to an inflammatory response.  These can often resolve but can take up to six months.  This is for the very fine spider veins.
If they are larger, then another treatment might be necessary.  You mentioned treating with a laser.  Did you have a vein doppler study done?  If not and there are larger veins under the surface, you are more likely to keep getting new veins until the deeper problem is addressed.
I recommend seeing an experienced vein surgeon/ follow up with your treating doctor to discuss these issues.
Good luck to you,
Dr. Powell