There’s really no other way, as Cloudflare isn’t going to grant you access to someone else’s Cloudflare account even if you’ve taken over the domain/site.
But if you also want to use Cloudflare, then you can add the domain to your own Cloudflare account, update the nameservers at GoDaddy to a new pair Cloudflare will assign, and point the domain (from your Cloudflare DNS) to Hostinger.