What are our options if: we didn't collect enough at closing for the initial escrow deposit because the insurance premium increased due to increasing coverage without us knowing.

Do a short year analysis now or let it ride until the next annual analysis. If it is significant, you might want to get that knocked out now before the borrower has a large arrearage.

OK, thank you sir! Could we go back to the borrower and suggest/recommend making an extra payment to the escrow account (after explaining the situation)?

You can't prohibit your customer from voluntarily contributing extra to his or her escrow account, but if you want to ask the customer for additional funds, you should perform a short year analysis:

1024.17(c)(1)(ii) Throughout the life of an escrow account, the servicer may charge the borrower a monthly sum equal to one-twelfth (1/12) of the total annual escrow payments which the servicer reasonably anticipates paying from the account. In addition, the servicer may add an amount to maintain a cushion no greater than one-sixth (1/6) of the estimated total annual payments from the account. However, if a servicer determines through an escrow account analysis that there is a shortage or deficiency, the servicer may require the borrower to pay additional deposits to make up the shortage or eliminate the deficiency, subject to the limitations set forth in § 1024.17(f).

The analysis will determine the actual amount of the shortage or deficiency. There are also rules (see 1024.17(f)) for the ways you can ask the customer to make up the shortage or deficiency.