You have a multi-year project ahead of you. As noted in these articles from Extension, persistence is key to success. Here is management information from MSU---
"Field horsetail has a deeply positioned rhizome system that is comprised of creeping rhizomes and storage tubers. Rhizomes can reach up to six feet in depth in the soil profile. Rhizome fragments, as small as one inch, can produce new plants. Control options for field horsetail are limited due to its extensive underground rhizome system. As with any perennial weed, several years of persistent control measures are needed to ensure success. Casoron is an excellent herbicide product for horsetail control; however, due to its residual activity, it will severely injure desirable plants placed in the treated soil for up to one year after treatment. Other herbicide options include MCPA and 2,4-D. Roundup is not considered a good, long-term option for field horsetail control. However, Roundup allows planting of desirable plants right after treatment. Always check manufactures label for specific details. Mechanical control options include pulling of the shoots or removal of the infested soil, but due to its rhizome system, soil would need to be removed to a depth of at least six feet. If that is not possible, placing a geotextile fabric in a shallowly excavated area before backfilling with new soil would prevent rhizomes from entering the bed. Fabric must be placed on the bottom, as well as the sides of the hole."
This article lists several licensed applicator products available to professionals. It also outlines some mechanical methods to eventually kill horsetail:
As always, read and follow all label precautions and directions on any chemical you may use. Thank you for using our service.