In Modern English, according to the OED, access is defined, in this context, as "The power, opportunity, permission, or right to come near or into contact with someone or something; admittance; admission."
Entry, on the other hand, is defined as "The action of coming or going in," in this context (while the word does have a definition listed which is synonymous with access, that definition is considered obsolete).
In other words, to gain entry into the building means to get into the building. This can be done sneakily, or with permission, or even by brute force, barrelling your way in.
On the other hand, to gain access literally means that you have gained the power or "permission" to enter the building. You are being allowed into the building by the building's security (whether this security is a system of guards or a computer system), or you can reasonably get past the guard.
Obviously, this can be done in a number of ways, from hacking into a computer mainframe to access security keys, to pickpocketing a guard for literal keys, to dressing up and presenting yourself as a high-authority figure, to buying an admission ticket. However, at this point in time, if you have "gained access" to the building, you now have what can be essentially described as an unlocked and unblocked pathway into the building.
If you get in, you have gained entry.
If you are allowed in, you have found a way you can get in easily, or if you have gotten in and you pose little risk of being removed, you have gained access.
The usual way of asking for assistance over the phone is
Could I speak to someone in accounts please.
I'd like to speak to someone in accounts please.
Accounts please.
I have a question about my account.
Being put through on a call is hold over from the days when there were central operators whom one had to speak to to make a connection, but the phrase can still be used today.
The lines were so busy, but finally I was able to get through to customer service.
The lines were so busy, but finally I was able to get someone in customer service.
The caller is not the one putting through the connection, but is the one who gets the connection.
Best Answer
As for me, Getting access would mean Obtaining credentials. You click this link to have an OTP or a token or a password sent to you. After you got access, you can use it to access the account itself.
Same with continuous: