Bank Loans for funding RE rentals

I recently got pre-approved for a conventional bank loan to purchase my 1st REI property . I was limited to the cost amount on a property I can purchase from their assessment from my financial history. However, for my 20% down payment would completely WIPE out my savings account and no money left for rehabs. I feel that this is a great risk because i would no longer have personal funds to cover any operational property expenses /vacancy / PM fees for tenant searches,etc. I know these expenses are analyzed into the expenses prior to making offers BUT its only on a monthly basis and just starting out saving for these operating expenses, there are NOT enough funds saved yet. Personal recourse is at risk.

Anyone have any suggestions I can pursue for purchasing my rental properties ,have rehab funds, and keep some savings as backup unforeseen expenses?  Thanks.