The short answer is no.
Ultimately, what a person needs to improve their FICO scores and build credit are three open credit cards (secured or unsecured) in good standing and one open installment loan in good standing such as a car, home, student, personal, share secured, or credit building loan. This combination is what the myFICO score theorists here have determined is what you need for optimal credit building and FICO score. You can have more CCs and more installment loans, however, this will not increase your FICO scores.
Next, (this is only important if you are attempting to purchase a large ticket item like a home or car) is paying in full all of the credit card balances each month, before the posting date, except one. This is called the All Zero Except One (AZEO) method. The one credit card you allow to post a balance needs to be less than 8.9% of the credit limit of the card. So using one card each month to buy lunch, letting it report and then paying in full will maximize FICO scoring. Keeping your utilization of your cards below 28.9% both individually and collectively will prevent you from incurring a FICO scoring penalty.
An installment loan will have its greatest impact on your FICO score when the amount owed is at 8.9% or less of the original amount owed which is usually in the final months before the loan is paid in full. If you don't have an installment loan you can check into Self Lender or a Share Secured Loan at a Credit Union.
Keep in mind, building credit is a marathon, not a sprint. It involves demonstrating to a potential creditor that you can handle credit responsibly. If you have open, active credit accounts that are being paid on time and pay all of your bills on time every time, apply for credit only when you actually need it and use credit cards sparingly maintaining low credit utilization, then you’re going to earn and maintain great credit scores. It would be impossible for you not to do so. This is the fastest way you can build good, solid credit.
Having an installment loan will definitely help your scores. I gain a nice bump when I opened an auto loan last year.
Your issue is you already have an installment loan with your mortgage, so an auto loan isn't really going to help your scores. Though having auto loans on your credit report with no negatives will make you more credit worthy for the next time you try for an auto loan as many places use an auto enhanced (or past auto loan emphasized) scoring model. So unless you really want a vehicle that requires you to get a loan, I wouldn't do it. The interest is just wasted money. I'd rather be debt free than maximize my credit scores.
