Should one stop taking contraceptive pill if they develope spider veins?

No. I have not seen a study that would recommend for you to stop contraceptives secondary to spider veins. Contraceptives can increase the risk of clots in some patient populations such as those over age 35 that smoke.
No. A pregnancy will cause probably more spider and varicose veins than contraceptive pills. You can use compression hosiery or compression socks and leggings, exercise and elevate your legs when you can to help slow down the vein disease and help prevent deep vein thrombosis.

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