Pain down the spinal cord after having a lumbar puncture?

LP complications. When spinal tap is done, if difficult, you might have local external bruising, swelling, and tenderness. Rarely, a local infection could occur. Typically, the usual reactions dissipate within a day or two. If symptoms are worsening, contact your doctor.
Unlikely the cord. Spinal taps are usually done in the lower lumbar spine. The spinal cord ends above that level. Midline pain after lumbar puncture is most likely from the lamina, interspinous ligament or muscle - relatively harmless.