The Athens General Index opened down about 4%, before recovering to be 1.4% lower. On Monday, the share index had dropped 16% on the first day of trading after a five-week shutdown.
Major banks were close to the maximum 30% declines they are allowed in one day, a limit that was hit on Monday.
Banks account for about 20% of the Athens index.
Falling bank shares were offset by gains in some non-financial shares. Among the shares rising were the gaming group OPAP and some tourism-related shares such as Aegean Airlines.
The main share index is down more than 50% from last year's peak.
During the height of the Greek debt crisis, the stock exchange was closed for five weeks as part of the programme of capital controls to prevent people taking euros out of the country.
Greek investors are still not allowed to use money in their bank accounts to trade on the stock exchange.
Greek bank shares have fallen sharply for a second day in Athens, but non-financial firms have fared better.