I have big database with user and passwords in sha256 hash. Now I write new version and I want to use sha256+salt. Is there a way to convert same passwords with sha256 to sha256+salt and have no trouble with login?
The way to salt and hash a password is to take the plaintext password, add the salt to it and THEN hash it. When you have an existing password database already sha256-hashed you don't have the plaintext passwords, and you can't easily get them back in plaintext once hashed (which is a good thing).
What you could do instead would be to take the passwords in their current form, sha256 hashed, add the salt and then hash them a second time - or better: many times, with better hashing algorithms.
The function to verify the password would then repeat those steps to check that the password is correct. Assuming the second hash is just sha256-hashing once to make the example clearer, though it's not sufficiently secure:
If you really want to avoid working on top of your existing hash you could create a new table of users where you process passwords in the new way from the beginning, and then gradually migrate user's passwords over to the new table as they log in with their plaintext passwords, and remove them from the old table.
A third solution could be to deactivate all the old accounts and require them to change their passwords before they can sign in again, via fx. e-mailing them a link to change their passwords.
Makes sense?
That said you will get more qualified answers at https://security.stackexchange.com . For instance I just found this question on why salt->sha256 hashing once is insufficiently secure, and another one here on how to process passwords for more secure storage.