"To continue the contract" means to extend or renew an existing agreement or arrangement with someone for a further period of time, typically in business or legal contexts. Full definition
Upon your passing, he / she will have the option to continue the contract in his / her name until the benefit has been paid out. (annuities.blueprintincome.com)
50 state comparison: Teacher tenure / continuing contract policies. (kappanonline.org)
A chance to continue the contract into a permanent role. (careerbuilder.com)