The museum, at Lambeth Palace in London, is now 30 years old and Woodward, appointed in 2007, wants feedback on how to develop it. He has co-opted Capital Gardens chief executive Colin Campbell-Preston and attracted industry figures such as Colin Squire in preparation for an industry-based ideas-producing event this year.
Woodward says the museum is now a hotbed of debate on garden design issues. And a new £200,000 mezzanine floor will provide space for the artefacts collected over the years. But he has now turned to the industry for help in deciding what are the most important parts of garden history of recent years.
"This is not opportunistic," he says. "We're asking what people want from the museum. I find it fascinating - architectural ideas change every two or three years and art every year, but gardens change every month. That's exciting."
He is floating ideas about displaying "objects of the year" - books, tools or plants. In 2006, perhaps the most distinctive image could have been the water butt; in 2007, Carol Klein's Grow Your Own Veg book. Be it the Flymo from the 1970s, Crocs shoes from 2005, the water sprinkler or the Bokashi in 2007, Woodward wants to know what the industry thinks. He is thinking of creating a shortlist for each year and organising a vote to "make sense of rapidly changing horticulture".
He adds: "The cultural history of horticulture and what it produces is interesting. It's how we look back through awareness of what people are concerned about right now."
Woodward's scope is wide at the museum. He is involved in landscape design, events and the Heritage Lottery Fund, of which he is a trustee.
He recently visited Lowther Castle in Cumbria, roofless since the 1930s and looking for lottery cash. Museum adviser and garden designer Dan Pearson won a competition for the garden revamp. Woodward says: "It's a lost garden, neglected since the 1950s. Dan is planning to unfold the layers. It's a steep and compelling landscape but it's a very competitive world for funding."
At the RHS Hampton Court Palace Flower Show, the museum and the National Council for the Conservation of Plants & Gardens, both founded 30 years ago, will show artefacts the public find in their sheds from the 1970s. It's all a new outlook for the institution.
"The museum is one of those places. (Founder) Rosemary Nicholson was inspirational. She worked six days a week and was a powerful, effective character. But what often happens afterwards is a loss of momentum. We have a good collection but most people can't see it."
He adds that while UK parks design is moribund, the museum is doing well on the contemporary design side. "(Landscape designer) Kim Wilkie's view of garden history is different to anyone's 30 years ago when the museum was set up. People wanted a space to meet and talk. Designers say they don't know what other designers are up to but they love hearing about Kim's latest projects."
The forum is the regular Vista discussions, which are over-subscribed, suggesting a gap in the market for gardening debate. This has been seen at the RHS and Oxford Literary Festival this year, as gardeners battle to be recognised as artists. "The Arts Council doesn't count gardening as art - garden design can only be art if it is by artists. Vista suggests gardening should be taken more seriously and not just as outdoor DIY." He also mentions that real debate is important: "At the RHS you can't have a debate, with 400 people all on one side."
Woodward also recognises that "we're always going to be tiny compared to the RHS" but is aware his evolving space suits many people better than the RHS's nearby halls, which are increasingly focused on non-gardening events.
Woodward asks: "What can you do in a building that you can't do in a visit to the garden? An exhibition can give you a different view and understanding."
Exhibitions will change three times a year and will be on the new 100sq m mezzanine floor from October. They will feature historical designers and contemporary retrospectives. The first is about a well-known living garden designer whom Woodward can't yet name, although he lets slip it will include revelations of her musical and artistic inspiration.
"We're inventing a genre. Everyone knows about art exhibitions. This is a garden designer exhibition. The puzzling thing about garden designers is their ambivalence about being in the spotlight compared with painters and architects. Christopher Bradley-Hole, for instance, hates talking about himself, and Dan Pearson is rather shy."
But would he put on a popularist Diarmuid Gavin exhibition? He says: "We're doing one on the 1970s and John Brookes. He would say gardening began in 1963. Before that we were growing veg in mud. He changed that for the garden to become a place where you could enjoy yourself. That exhibition is aimed at the mainstream market. The museum can define the long-term values of a changing world."
1987-1990: Studies as an art historian at Cambridge
1995-2000: Curator, Sir John Soane's Museum
2000-2005: Director, Holburne Museum of Art, Bath
2001: Becomes interested in landscape and gardens when writing his first book In Ruins (Chatto & Windus, 2001)
2006: Director, Museum of Garden History
2007 to date: Trustee, Heritage Lottery Fund.