You do not need to apply for another ESTA, because ESTA is generally valid for two years. Applying for another ESTA will not help. ESTA is simply an authorization to travel to the border; the actual terms of your admission are governed by the Visa Waiver Program (VWP), under which each entry is limited to 90 days.
In theory, it's possible to leave the US (or, better, to travel outside North America and the Caribbean) and immediately re-enter for another 90 days, but the chance of your plans being frowned upon by an immigration officer are very high. If you try to re-enter the US for 90 days immediately after having spent 90 days there, you are likely to be refused entry, but the longer you stay away, the more likely you are to be able to return for the full 90 days.
You can also apply for a B-2 visa, which will allow you to stay for six months at a time. The application fee is $160, and the visa is generally valid for 5 or 10 years, but this depends on the country whose passport you hold. If you are going to do this, you should be able to answer the following three questions clearly and convincingly.
- Why are you staying in the US for longer than 90 days?
- How are you going to support yourself while you are in the US?
- Why should the visa officer believe that you will leave the US?
Your answers should be supported by evidence, and none of the answers should indicate that you are going to violate any conditions of visitor status. For example, your application cannot indicate that you plan to support yourself by working in the US.
The fact that you are a recent graduate will make refusal of your visa more likely. It might be safer for you to change your travel plans somewhat so that less of your time is spent in the US.
Another important factor to consider is that if your visa is denied, you will "most likely" not be able to use the VWP in the future. Thanks to Zach Lipton for pointing this out in a comment. The linked page begins:
If you were previously denied a visa, or previously refused entry to the United States, or previously removed from the U.S., your ESTA application will most likely be denied.
If you were allowed to board your carrier, you may be subject to additional processing upon arrival at U.S. ports of entry, and may be denied admission to the U.S. Applicants who are uncertain of whether they qualify for travel under the Visa Waiver Program (VWP) are encouraged to apply for ESTA authorization early, to allow time to apply for a visa, if needed.
As to the return ticket requirement, it is okay if the return date is longer than 90 days after you arrive. The specific requirement is at 8 CFR 217.2:
Round trip ticket means any return trip transportation ticket in the name of an arriving Visa Waiver Pilot Program applicant on a participating carrier valid for at least 1 year, electronic ticket record, airline employee passes indicating return passage, individual vouchers for return passage, group vouchers for return passage for charter flights, and military travel orders which include military dependents for return to duty stations outside the United States on U.S. military flights. A period of validity of 1 year need not be reflected on the ticket itself, provided that the carrier agrees that it will honor the return portion of the ticket at any time, as provided in Form I-775, Visa Waiver Pilot Program Agreement.
But once again you should keep in mind that the immigration officer has wide discretion to deny entry. He or she can (and probably will) take the return date of your ticket into account in assessing your plans. If you plan to leave the US and return immediately to extend your stay, the officer is likely to deny entry. On the other hand, if you have a clear plan to spend a lot of time outside the US between your arrival and your return flight, you are more likely to be admitted.