If you're finding that your phone is only storing a few days worth of emails, you can change this by following the below guide
- Open the settings app on your phone and scroll down to 'Passwords and Accounts'
- From there, you want to select the account that has the email address in question. In the example below you can see it's named 'Work email' but this may be different for you.
You can always check what address it is by tapping on the account.
- Tap on 'Mail Days to Sync', and from there you can select how long you'd like to store emails on your phone for. The example below is showing as having 1 month selected.