There are lots of different levels of contact, but generally alternate weekends is probably the norm where there are no welfare concerns, and then maybe some time during the week if practical. In your position, you're actually not far off, and in fact you have contact every weekend, so you could perhaps try for 2 nights sleepover on the alternate weekends, but bear in mind that your ex is also entitled to have weekend time. Your first step is to see if you can agree a bit extra contact, and if not, then try mediation.
My son sees his son on 2 week rotation, Tuesday after school 12 till 6 ( he is in court process now trying to change this to an overnight) then Friday from school overnights all weekend, he takes him to school Monday morning. Then Thursday after school overnight, back to school Friday morning. He achieves this by working a zero hours contract, but now looking for permanent work, hopefully new employer will fit around this times, his son is 4.
The courts would expect you to try and sort this between you, or by mediation. If you want something drawn up official, I imagine a solicitor could do this for your ex to sign.