Hi Maurice,
From the years where I have done SEO, it does help to keep the same page and link structure etc.
Make sure you get Google to recrawl your site asap to reduce any lag times.
There will often be small dips in your SEO, but Google tends to trust your domain over everything else, so as long as you are improving your site it will always have a long term effect in your SEO.
I re-launched a friends hair salon website. She fell off page 1 for “hairdresser joondalup” (her suburb), but after 4 weeks she was back to no.1