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

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