The new chief executive of gay rights group Stonewall is to meet leading members of the UK's transgender community to see how the organisation can best promote rights for all members of the LGBT community.

Speaking to the Guardian on the first day of her new job, Ruth Hunt said Stonewall, which does not campaign or lobby on transgender issues but works with transgender groups, would be opening conversations with the trans community at a meeting at the end of August.

"We are at quite an important tipping point in terms of trans equality, and we are looking at how we can best support and maximise that tipping point," she said.

Hunt added it was important for Stonewall to be led by the needs and desires of the transgender community. "Any change needs to be led by the trans community … we are very open to taking whatever direction will be in the best interest of [that] community," she said.

After a decade of seismic change in gay rights – which has seen the introduction of civil partnerships, equality legislation and gay marriage – one of the biggest challenges facing Stonewall was a "creeping complacency", said Hunt.

"Our motto here is a lot has been done – but there is a lot to do," she said. "Some people think because we have achieved significant legislative goals, the job is done but changing hearts and minds is a much longer job."

Hunt, who is a Catholic, said there were still many isolated gay people, including those with faith, throughout the country who needed support. "Some have gained more from these legislative changes than others," she said. "People living outside big cities people belonging to faith groups – I have been speaking to a young woman who is a committed Muslim and gay, and she can't imagine speaking to her parents, never mind meeting a partner – there is still a lot to be done."

Hunt has been acting chief executive of Stonewall since Ben Summerskill stepped down in February. She has been at the organisation for nine years, starting as a policy officer and working her way through the ranks to become head of policy and research, director of public affairs and deputy chief executive.

The second woman to lead the organisation – the first was Angela Mason, who successfully lobbied for amendments to the 2002 Adoption and Children bill, which treated lesbian and gay couples in the same way as heterosexuals – she said having a woman at the helm again would help reflect the diversity of the lesbian, gay and bisexual community.

"For some people gay means gay man, and I think already we are seeing a level of support from women, who perhaps haven't been as involved in the past," she said.

Stonewall would continue to look for ways to support lesbian, gay and bisexual rights groups around the world, Hunt said, adding: "I think it is important that we don't try to impose an idea of of what gay rights should look like in each country, but we support groups working in those countries. It takes time, it doesn't lend itself well to a hashtag but it's about saying, 'How can we try and help you do what we did?'"

She said she was "hugely excited" to be taking up the role. "Stonewall is such an important organisation doing such incredible work," she said. "I'm very proud of the work we do, and how we do it."

Jan Gooding, Stonewall's chair of trustees, said: "Ruth is an outstanding choice for Stonewall – a bold leader with a naturally collaborative approach and a clear vision. Stonewall is 25 this year and at an exciting time in its evolution.

"We are justifiably proud of our achievements in equal rights for lesbian, gay and bisexual people in the UK and now the charity is ready to begin a new chapter. Ruth has amply demonstrated that she has the right skills and experience to take us forward, helping us achieve real and lasting social change."