There are several routes into working in the UK, however Graduate route, Start Up, Skilled Worker route and Health and Social Care Worker routes are most popular with students.
You can find all options listed, with detailed links at https://www.gov.uk/browse/visas-immigration/work-visas. These include some routes that do not require a sponsorship or those for working for an overseas employer. There are some Temporary Work options that might be of interest.
Please be aware that once you switch out of the Student route before completing your course, you will no longer be eligible for the Graduate route visa on completion of the course.
The International Experience Advisers specialise in student immigration applications are therefore unable to offer advice for non-study related routes. If you are instructing a solicitor for help, make sure they are a member of ILPA (Immigration Law Practitioners Association) - https://ilpa.org.uk/
Please note that it is against the law in the UK to charge someone to find them a job. Genuine agencies get paid by the employer, not the employee/potential employee. Job agencies and websites should not ask you to pay for help in applying for or getting work. The UK Government has guidance on what you can expect from an employer when you look for a job. See Your rights and safety when looking for a job. Do not take a job if you have doubts about the employer and how they will treat you.