The GII is one of the most advanced indices to show a picture of innovation in society.
The GII aims to improve the measurement of innovation in order to provide a more complete picture of innovation ecosystems across the globe. It explores new metrics regularly to reflect the changing nature of innovation and the increasingly sprawling field of new (big data) innovation indicators.
the GII has also made a special effort to cover creativity and creative outputs, taking a fresh view of the previously siloed approach to innovation versus creativity.
So to improve ranking you just need to revise GII indicators. Some of them can be improved easily (e.g. government R&D spending), others are more complicated to improve (e.g. output indicators, like Domestic industry diversification, high tech exports etc.) But also you should note, that some GII is dynamic and new indicators might replace some old ones. More detail is here
Why would you want to improve a country's ranking in the Global Innovation Index?
Why not to improve the innovation performance of a country, especially the favourable socio-economic impacts of innovation?
Considering any ranking can be easily misleading. It is more pertinent, more useful to have a good understanding of the national innovation system of a country (and depending on the country, also the regional and sectoral innovation systems inside that country).
For the pitfalls of GII (and the European Innovation Scoreboard), and more generally, that of composite indicators, see, e.g.
Havas (2019): Measurement of innovation: the use and misuse of indicators and scoreboards (available here, at RG)