Besides half-day and day rates, you can do what most services firms and design agencies do: Estimate based on hours of actual design work.
This lets you get a bit more specific, exercise control over rounds of revisions (by scheduling the project based on how long it will take), and it lets the client dial up or down the number of revision cycles or complexity to fit their budget. An obvious other question is the schedule, and how much design work is possible in that timeframe.
If this is done, I also pad the estimate not to gouge, but to reflect risk. If this is a known or repeat client, the risk may be zero. If this is a new client and there are a lot of stakeholders, I may assess a 10%-20% risk factor to my estimate. If there's a lot of meetings, I'll add 5% meeting overhead, too, if I'm only accounting for hours of design work. Finally, if it's a juicy gig or a good cause, I'll discount, while making sure it's a conscious decision to win the gig. Loss-leader projects are rare, but thin-margin projects to win the work are a little more common.
I always start with a verbal range to start the conversation, explaining what the biggest cost drivers are, which are usually the amount of design time itself as well as the number of revision rounds. (Actually, I cut to the chase first, and ask if they have a specific budget in mind that I should be working backwards from. Most of the time they decline, but then it's not your bad if your estimate takes them by surprise.) Then I gauge the reaction, and negotiations ensue. I always go in with a fair number up front, but I do go into negotiations having a bottom number in mind, at which point I'll feel good about walking away.
The biggest unspoken advice in the work-for-hire world is that gigs are 50% design and 50% expectations management. That latter 50% is vital to establish from the get-go, as it cements your position as the expert and what the client is expecting...this is also the main driver in your ability to keep your work schedule sane.