In September, the government announced a 1.25 percentage point increase in National Insurance with the money split on health and social care. This will come into effect in April next year.
A new health and social care levy - offering more funding - will come into force the following year.
But Hampshire County Council wants the cash quicker.
It comes as a county council report revealed there will be a £14m social care budget gap by 2023/24, with no long-term funding solution.
Cllr Liz Fairhurst, executive member for adult services and public health, said: 'It is a worrying time and that's no different for other councils up and down the country.
'There will be money coming from the social care levy but I honestly think we need it before 2023.
'That is the message we're trying to get to the government - we are lobbying them as much as we can.'
The council is expecting a constant increase in adult social care demand, together with rising costs and labour shortages.
Hampshire County Council has been running a Call To Care campaign, in a bid to drive recruitment in the sector.
Cllr Fairhurst, who described the scheme as having 'some green shoots', says more work is being done behind the scenes.
She said: 'I'm not sure what the permanent solution is, and there is no estimate for how long we can keep going on like this; in the meantime there's plenty being done to support the social care sector.
'We are working much more closely with the NHS and talking about integration, working under one system, which would make a huge difference.
'We are preparing as best we can to meet the expected demand. The county council has an extra care programme and making better use of technology.
'But it's not just a question of money, it's about finding people to do the jobs too.'