Hi,
I have read your exact requirement in which you have mentioned that you want your existing site should be more speedup.
Please share the URL of the site so I can check the current performance and issues on gtmetrix and Google Page speed.
I will optimize the complete site and server so the load time wil be reduce and will achieve a very good score for mobile and desktop both.
Query: Do you have shared Hosting or Dedicated server?
Following some steps so we can increase the speed of site:
1. compression
2. caching (in caching ...pagespeed ,varnish ,redis if root detail available)
3. levarage browser caching
4. optimize images minify css and js
5. add expires headers
6. database optimisation etc
Regards
Karuna S