Could you get a better savings account from a bank you've never heard of?

After months of falling interest triggered by the start of the coronavirus pandemic, there are now signs that the savings market is slowly improving, with some average rates creeping upwards.

Fixed-term savings account rates are looking particularly encouraging, with average rates for one-year fixes rising to 0.48% in June 2021, up from 0.44% in May.

Similarly, the average long-term fixed savings account (ie those with terms of 18 months or more) now pays 0.72%, up from 0.66% the previous month, according to data from Moneyfacts.

But how much of this interest rates boost is being driven by the big high street banks?

Here, Which? reveals the providers currently offering the top rates for savings accounts and cash Isas - and you won't find any of them on your local high street.

Best savings and cash Isa rates

The table below sets out the top rates for restriction-free fixed-term and instant-access savings accounts and cash Isas, by order of term.

_{*Expected profit rate. Source: Moneyfacts. Correct as of 4 June 2021, but rates are subject to change.}

As the graph shows, the top rates on the market tend to come from smaller providers - some of which, such as Atom Bank and Zopa, are considered to be 'challenger banks' as they only operate via mobile or online.

Others are unlikely to be on your local high street. United Bank UK, for instance, has just three UK branches, located in London, Birmingham and Manchester. Similarly, there are just seven branches for Punjab National Bank: four in London and three in Leicester, Birmingham and Wolverhampton.

As is usually the case, it's tricky for those with savings of less than £1,000 to secure a top-rate account, as this is the most common amount required for a minimum initial deposit. However, it's encouraging to see the two top instant-access accounts allowing savings from £1.

Note that the account from the Bank of London & The Middle East is a Sharia-compliant product, and so offers an expected profit rate (EPR) rather than an annual equivalent rate of interest (AER). This means that the advertised rate isn't guaranteed, but we've never heard of an instance in the UK where a bank has paid less than its advertised rate.

• Find out more:how to find the best savings account

You might need to act quickly to get a top rate

Banks or building societies use savers' deposits to fund their lending, as it tends to be cheaper than borrowing the money themselves. Given that the property market is currently experiencing a boom, it makes sense that some providers want to encourage more savings as it means they can fund more mortgages.

However, given that many of these top rates are being offered by smaller providers, savers may have to move quickly before it gets withdrawn. That's because smaller providers will need to raise less cash in order to reach their lending targets; if eager savers have funded what they need within a few days, then it's likely that either the product will get withdrawn or the interest rate will be reduced.

• Find out more:how to find the best cash Isa

Top-rate notice accounts

If you don't want to commit to fully locking your money away, a notice account could be a good savings option.

As the name suggests, you have to give your provider a set period of notice before it will release your money, but you're generally able to make as many withdrawals as you like. As with instant-access accounts, the rates could change at any time.

Notice terms can range from seven days to over a year, but the most common terms are 30, 60 and 90 days. Some of the market-leading rates we found are:

• Shawbrook Bank 120-Day Notice Personal Account, paying 0.72% AER, with a minimum initial deposit of £1,000.
• Allica Bank 95-Day Notice Account, paying 0.7% AER. It also requires £1,000 as a minimum initial deposit.
• Paragon Bank 120-Day Notice Account, which pays 0.7% AER and requires a £500 minimum initial deposit.

What is the state of the savings market?

While it's true that we're seeing a spike in competition for top rates and some average rates are also on the up, it's still a little early for savers to start celebrating.

The graph below shows the average rates for savings accounts and cash Isas, covering instant-access, one-year fixed and long-term fixed accounts, using data from Moneyfacts.

As the graph shows, there are still several categories where average rates have remained the same for several months - this is the case for instant-access savings accounts. And, while it's positive that these rates are no longer falling, they are still a long way from where they were in July 2020, when rates had already started to tumble.

Indeed, even though average one-year fixed-term savings account rates have increased in June 2021, they're still lower than they were at the start of the year, and 0.22% behind the July 2020 rate.

You can search through hundreds of savings accounts with Which? Money Compare.

Which? Limited is an Introducer Appointed Representative of Which? Financial Services Limited, which is authorised and regulated by the Financial Conduct Authority (FRN 527029). Which? Money Compare is a trading name of Which? Financial Services Limited.