if the posts are fixed into stirrups and the rafters all bolted together the most effective way would be to dismantle the whole structure and replace the posts with taller posts...the existing tall posts might be long enough to recycle into the new short posts but you may find that steel posts (powder coated to colour coordinate?) could be more suitable than timber and reuse the rafters and shade cloth but we've found that black shade cloth is much more cooling than other colours and you may also need vertical shade cloth for a privacy screen from neighbours windows when the pergola is raised higher

When you say too low, what is the problem the height is causing?

Change the pergola to a Gabled roof this will not only look good but give the extra height.

I love Siruskeys idea !

I must admit , if it was me , I would have been more inclined to get some post extensions ( wood the same 100 x 100 or whatever it is ) , unbolt the top , get some galv brackets made with a 'hook' on half of them , bolt the old and new together using the brackets , then do everything in charcoal .

Of course , these new brackets ( flat steel say 30-40mm wide and 6-8mm thick ) will look a bit funny , and the charcoal stain won't take the same , so they'll always look a bit out of place . But thats why you get the engineers to put a 'hook' on them before you send to the galvanisers -- a bit of 8mm solid rod coming 120mm or so out , with a couple of small bends , one down , one up .

Yes , hang hanging plants on them -- takes the focus away from the poles , the extensions , adds some colour and 'life' . But I'm a lot more rip than sirus haha !

Oh , and I'd paint or tile the deck , probably charcoal to match the windows , or terracota to suit the style , but even off-white or grey ( if tiling ) would work . The brown less so .

You know me Pottsey, keep it simple works for me

The gabled idea looks great, better than the flat.
Interesting comment on the black shade cloth colour too, a good hint for future reference.

Retractable blinds on both sides of the gable made from black shade style cloth would work to control different times of year. Two recently renovated houses near us have used these and they are a great feature

I would never have thought of gable but really like it! Interestingly we started with a black retractable shade cloth but it radiated too much heat so replaced it with a lighter colour which is better.

Good to know, the houses near me already had alfresco areas and the black retractable blinds were automatic side blinds and not over head and look excellent. I think the gable will look great on your house remembering to try and keep the same pitch as your existing gable roof if it fits looking forward to seeing what you do.