Do I need to find a visa sponsor to stay and work in the UK after my studies for a Master's/PhD?

No, you can legally live and work in the UK right after your studies without finding an employer sponsor first, thanks to the Graduate Visa route. There is no job offer or minimum salary requirement to qualify. You just need to apply from inside the UK before your current Student Visa expires.

Graduate Visa Duration by Qualification

Rules to Keep in Mind

• You must apply from within the UK; you cannot leave and apply from your home country
• You can apply once your university reports your course completion to the Home Office, without waiting for your graduation ceremony
• The Graduate Visa is a one time opportunity and cannot be extended
• Near its expiry, you will need an employer with a valid sponsor licence to move onto a Skilled Worker Visa, and switching from a Graduate Visa often qualifies you for easier "new entrant" salary thresholds

My Advice

Use your Graduate Visa years to build real UK work experience in your field rather than waiting passively for the perfect sponsored offer, since strong experience makes employers far more willing to sponsor you later. Start researching companies with an active sponsor licence about a year before your Graduate Visa expires, so the transition to a Skilled Worker Visa is not a last minute scramble.