The once claiming the child MAY qualify to file as Head of Household HoH). Two rules apply: you must be claiming a related dependent and you must be the one providing more that half the household support. This usually means that only the higher earning parent can be HoH. So, being able to claim HoH factors into who claims the child. But, a bigger factor, usually, is who gets the bigger Earned Income Credit.
If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice. It may be worthwhile to prepare trial returns, both ways, to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.
