Not true. You do not 100% need an agent to open a bank account. Some banks will permit foreigners to open bank accounts.Also it's not change of visa type because 60day exemption is not a visa.
A long stay visa from your own country will be OA requiring health insurance . Equivalent of 800,000 baht in your home account will satisfy the fund aspect for the first year. However onward travel is simply a matter of buying an online bus ticket out of the country. Arrival on 60 day visa exempt and then apply for 90 day extension 3 weeks before visa exempt period expires. Ensure funds are deposited for the requisite time - it can be done. The 90 day extension will start from the end of the 60 days exempt period. As soon as the 90day visa is issued, get a resident's certificate from the immigration department issuing the visa. Take it to the bank to open an account.