When Asking for Amicus Briefs, Give Plenty of Lead Time
Parties sometimes wait until their brief is filed before asking for an amicus brief -- but that usually leaves just days or at most a few weeks for the amicus to write the brief.
If you're interested in having someone file an amicus brief, you should give a lot more lead time than that. In federal courts of appeals, amicus briefs are due 7 days after the brief of the party they are supporting; even for briefs supporting petitions for certiorari filed at the Supreme Court, the due date is generally 30 days after the petition; in many state courts, the timing is comparably tight.
That's very little time for someone to review the papers, decide whether to file an amicus brief, and then write and finalize the brief. And it's even harder when the request is for a brief to be filed by an advocacy group, which often has its own time-consuming institutional review structure.
So if you're thinking about trying to get amici in a case before an appellate court (including a supreme court), you need to ask them well before your briefs are filed. The best time to ask is as soon as you've decided to take up the decision below, so you can send the prospective amici a copy of the decision below, the briefing below, and a general sense of the arguments you'll be making before the appellate court. That gives the prospective amici months to do this, rather than weeks. A petition for certiorari, for instance, is usually due 90 days after the decision below, though even that can often be extended; if you ask prospective amici for help even 15 days after that decision, they'll have more than three months to prepare something (90+30-15 = 105 days).
Now it's true that some courts do allow amici to file motions for leave to extend the filing deadline. But few amici are likely to want to do that, and risk having their briefs rejected. (To be effective, such a motion would usually have to include a copy of the accompanying brief, so the brief would have to be drafted before the court decides whether to allow the late filing.)
I realize that, when one is occupied with writing one's own brief, it can be hard to turn one's attention to related matters, such as lining up amici. But asking for amicus help is pretty quick and easy, compared to many things in the litigation process. And it's much more likely to be successful if you give amici plenty of notice.