Opening a second Pillar 3a account is often a good idea when your first account's balance hits around CHF 50'000. Overall, whether you should stick with the same provider or choose another provider depends on your preferences.
Think about why you chose your existing provider in the first place, explore the options or incentives available to signing up elsewhere, and see what best aligns with your needs.
Importantly, you need to ensure that if you use the same provider, your accounts are actually registered as separate accounts (as opposed to sub-accounts or sub-investment strategies). This means you can still benefit from staggered withdrawals when you come to withdraw the funds later in life.
Opening a second account with the same provider
With everything 'under one roof' with the same provider, you will typically have easier management of multiple accounts. This could be in the form of only having to use one app on your phone, or only one login on your computer, to access all of your accounts.
As you have already decided upon a Pillar 3a provider, there are likely reasons for doing so (such as low-costs, large investment selection, etc.). Sticking with the same provider allows you to continue to benefit from the features that made you sign up in the first place.
Additionally, you can sometimes benefit from lower fees if you hold multiple accounts, or have a larger total amount of funds held, with the same provider.
Opening a second account with a different provider
Opting for a second provider could offer you different investment approaches, asset allocations, and fund choices. This means you can spread risk across multiple strategies rather than relying on a single one. Naturally, there may be overlaps in the underlying holdings, though this is important to check beforehand.
You will also be better protected against provider-specific risks. Relying on more than one provider means you are exposed to multiple internal policies, fee changes, and operational decisions, enabling you to freely move away from one if you do not like the direction in which their policy is going.
There is also the risk of any single institution going bankrupt. Whilst your money will likely be safe (though this is only guaranteed up to CHF 100'000 per Swiss bank if held in cash), one provider going bankrupt can result in a lot of administrative work to ensure your funds are kept in your name and moved to another provider of your choice.
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Please note that all content within this response has been prepared for information purposes only. This response does not constitute financial, legal or tax advice. Always ensure you speak to a regulated Financial Adviser before making any financial decisions.