The rates on offer from savings providers have improved from record lows seen in August.
But the shortening “shelf life” of some offers means savers need to be ready to pounce if they see an attractive deal, according to Moneyfacts.co.uk.
Average fixed rates for one-year and longer-term fixed bonds as well as Isas increased in September, with greatest uplifts seen on longer-term options, Moneyfacts said.
The average easy access Isa rate has increased from 0.32% in August to 0.35% in October.
Typical one-year fixed Isa rates have increased from 0.56% to 0.62% over the same period.
Fixed rates offered on longer term Isas with terms lasting more than 550 days have increased from 0.75% on average in August to 0.85% in October.
Rates on longer term bonds with terms of more than 550 days have gone up from 0.84% in August to 0.93% in October on average.
The average one-year fixed-rate bond has moved up from 0.63% in August to 0.68%.
There are also more deals to choose from compared with over the summer.
There are 1,531 savings deals including Isas on the market, the highest level since May.
Despite this, there are 322 fewer deals available than a year ago.
Deals are also hanging around for shorter time periods before providers tweak their ranges.
The average “shelf life” of a fixed-rate bond fell to 28 days from 36 last month, the lowest number of days since March 2009.
Rachel Springall, a finance expert at Moneyfacts, said: “Many consumers have amassed a substantial portion of disposable income over the UK lockdown and with economic uncertainty prevalent, it appears this cash is flowing into accounts that can be easily accessible, and providers are moving fast to cope with demand.”
She said rates had been pushed to the record lows in August, adding: “However, as the market remains very fluid, its rejuvenation may well be a slow and steady process and savings providers must adjust to demand within the space of a week or even days in some cases.
“Providers will have to review their savings range carefully and remain resilient in these uncertain times, and savers will need to act quickly to be in with a chance to acquire the best deals.”