You get Conscript companies (CS) two ways:
1. Certain cities give you CS points each turn(there is usually 1 city per state which gives them, and by hovering over it you can see how many it gives you). This is a small steady stream of CS points. You can add generals with the ?recruiter? tag to get more CS points. There are several USA generals who can do this, Banks, Burnside (I think), McClernand (I think), but only 1 CSA general (you get him in 1863 IIRC).
2. Political options - you can call for volunteers and/or draft. You can cycle between different options on the screen. So for example there is the default "call for volunteers", but you can also pay a $1,000 or $2,000 bounty to get more CS points. The same with the draft, you can either "partial draft" or "full draft". These options reset twice a year (so you can call for volunteers up to twice per year, or draft twice a year. This is where you get your main armies from. You will note that troop raising comes in waves, and this is due to the twice a year for draft/volunteer options.
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Remember - The beatings will continue until morale improves.