Squeezing better returns from your savings is almost always possible, and it's worth a regular review. Sometimes better interest can be earned by switching to an online only account or a short-ish term deposit.
There are other tricks:

Notice saver vs term deposit
I'm a fan of notice saver accounts, which a few banks offer. Notice savers don't have fixed end dates. when you have to go through the hassle of reinvesting. Instead, you give notice when you need the money. The notice periods are often 30 days or 90 days, depending on the account you've chosen. One useful advantage is you can top them up a few dollars at a time when you have the funds. Notice saver interest rates often beat term deposits.

Bonus savers
These accounts pay a monthly bonus or offer prizes if you don't make withdrawals. The base interest rate, however, is low.
In order to qualify for the monthly bonus you must fulfil certain conditions. That might be not making withdrawals in a given month, or the overall balance must grow by a set amount, such as $50, every month.

Notice periods and minimum deposits
Often the longer you are willing to tie your money up for the higher interest rate.
At Westpac now, for example, you'll receive 1 per cent interest per annum for a one month term deposit, up to 3.35 per cent for nine months, and 4.10 per cent for five years.
Likewise, the larger your deposit the higher return you get. At BNZ, for example, the 12 month rate is 0.75 per cent on a $2000 term deposit, but 3.15 per cent for $20,000 or more over the same period.

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Smaller banks, societies, credit unions
Overseas banks Kookmin and ICBC online have the same credit rating as Kiwibank and often offer good rates. Credit unions support their customers and communities and offer competitive rates. I'm a great believer in not-for-profit financial institutions.

Laddering
If you have larger deposits but don't want your money tied up seemingly indefinitely, you can ladder your term deposits. For example, every three months you invest a quarter of your money into a 12-month term deposit.
Over time this means that every three months you get a quarter back, but are receiving the better 12-month interest rate overall.

Peer to Peer lending

Harmoney, Lending Crowd, Squirrel Money and others allow lending direct to borrowers and take the majority of the profit that the bank or finance company usually keeps. Returns of 11 per cent per annum or more after fees and failed loans are possible. Always research the risks.

There are also traditional finance companies such as UDC and Fisher & Paykel Finance still in business.

Revolving credit
Using short-term savings to pay down your home loan returns the equivalent of your mortgage interest rate, but tax free. Make sure you have the type of mortgage you can overpay, such as a revolving credit mortgage, but beware of using it as a piggy bank.

Managed funds

Income-focused managed funds spread your money across low-risk investments, should return more than a term deposit after fees and the money can be withdrawn immediately.

KiwiSaver
If you're saving to buy your first home or for retirement, KiwiSaver is a pretty good place to put your money. It can't be touched, and generally has low-ish fees, the member tax credit and employer contributions.
As I've said before, make sure you're in the right fund.

I can't stress enough that the higher the interest rate the greater the risk you face of losing your money, no matter what the fancy marketing tells you.
I still recoil in horror at the reader who emailed me to say he was putting all his money into peer-to-peer lending.

So before depositing your life savings, check the company's credit rating, if it has one, with Standard & Poor's or Moody's. Be aware that unlike school, a B rating for a company isn't flash.
Anything less than an A- comes with a higher element of risk than the high street banks.

Finally, if you really do want the best returns you need to review your savings regularly, shop around, keep your money on the move as interest rates rise and fall, and reinvest rapidly.

*Average per annum over 10 years.