The call for highest and best can feel like throwing stuff blindly at a wall in the hopes it will stick. It's best to remain calm, review the comps (again) and decide on what this house is worth it to you. Make an offer that will leave you with no regrets if yours turns out not to be the highest and best. That really is the secret for any bidding war! Don't bid out of your comfort zone and keep in mind, if it doesn't work out, it's not the right house.
Could you be offering much more than the 2nd highest and best offer?
Possibly. One way around that is to include an escalation clause. You offer a price and agree to escalate up to a certain amount in steps. If you offer 300K for example but are willing to go as high as 310K, then you say you'll up any other offer by $500 or $1,000 until your max is reached. Again .. you decide what that max is. You will still have an appraisal contingency in place to make sure your are not purchasing the house for more than its market value. The comps should still support the price unless you are willing to pay cash over the appraisal value.
You can also tweak your other offering terms a bit, price is just one - albeit the most important - factor. Perhaps your agent can find out which closing date would be ideal for the seller. Maybe you can reduce the closing help you are asking. You can make sure your loan preapproval is as strong as it could possibly be and you could also increase your earnest money deposit.
When the seller calls for highest and best, don't panic! Decide what the house is worth it to you and put your best foot forward.