What do you recommend to do when your savings account balance exceeds the cap for higher interest rate (i.e. above 70000 for OCBC 360)?

Agreed with Kenneth. There are a number of non-hurdle, yet decently yielding savings account out there.

The first one I'd definitely recommend is the SCB Jumpstart account. It pays 2% p.a. interest for balances up to $20k. One condition is that you have to be between 18 - 26 years old. https://www.sc.com/sg/save/savings-accounts/jum...

Alternatively, the CIMB FastSaver is another attractive savings account that you may want to explore (1% p.a. for first $50k of deposit)

https://www.cimbbank.com.sg/en/personal/product...

If all else fails, the Singapore Savings Bond is always there for you. Latest issuance yields 1.71% p.a. if held over ten years. One good thing about SSB is that there is no early-redemption penalty unlike Fixed Deposits.

https://www.mas.gov.sg/bonds-and-bills/Singapor...

If you have spare funds above all of these, you may also want to explore Money Market Funds. Fundsupermart has a relatively low-cost offering for this:

https://secure.fundsupermart.com/fsm/funds/fact...

Err... You can open more high interest savings accounts? Hahha.

UOB One account doesn't require salary input I believe.

But yes, you should also consider how much liquidity you need. Even if you are highly conservative, there are probably instruments out there that you can leverage to help your savings grow. CPF? SSB? FD?
You should also take into consideration your age, what you might need the money for in your forseeable future, how long you can leave it aside, etc