The single best way to improve your odds of winning a grant is to build a personal relationship with the sponsor-point-of-contact - i.e. the person working for the grant awarding agency who puts out solicitations for the grant.
As a startup working on a challenging technical problem, you can always browse through the open solicitations during the times when solicitations are open and agencies are accepting application. However, cold-applying to any given solicitation without talking to the team and leader who released the solicitation brings a very low probability of success. Winning an SBIR grant is like raising money in any other environment: It is a relationship game. You need to find out who within which relevant agencies are interested in the general area you are working in. Then you need to find them and have a conversation with them to understand what their needs and research interests are.
The most successful SBIR awardees build strong relationships with the sponsors, so that they understand the latter's interests completely. By engaging them in dialog well before the solicitations are released to the public, teams have the opportunity to understand what is needed and also provide input and influence into how the sponsor thinks about, and then writes, the solicitation. Then when the solicitations are released, they are written in such a way as to favor the applicant who helped craft the solicitation. Their odds of winning that award is vastly improved compared with straight up cold applications.