Would the fact that their hands were tied with regards to mitigating damages mean they could recover for more than just the initial damages?
That depends on the terms and the sequence of the contracts involved.
The terms of a contract might reflect the extent to which parties are willing to take risks. This is typically palpable in the provisions or contractual remedies in the event of breach.
The sequence in which the plaintiff entered his contracts as well as the timing of a breach (as known to the plaintiff) are also relevant because they reflect plaintiff's awareness of his obligations pursuant to all other hitherto enforceable contracts.
Suppose A enters two separate contracts C-AB and C-AC with B and C, respectively. At some point B breaches C-AB, and A sues B for breach of contract.
Defendant B will argue that plaintiff A was knowingly bearing risks in a way akin to Restatement (Second) of Contracts at § 154(a)-(b) when entering either or both contracts. Thus, the terms of C-AB would have to supersede the presumption that A agreed to an allocation --even if implicit-- of risks that is unfavorable to him.
But if B breaches the contract prior to the formation of C-AC, then A shot himself in the foot by allowing in C-AC the constraints about mitigation of damages.
If the aforementioned aspects are inconclusive, then the matter would have to be decided on equitable grounds and what is reasonable under the circumstances (see Restatement at § 154(c)).