First, let me congratulate you on this great idea of a 'smile & greet' day. It's a good way to institutionalize the practice of smiling and greeting. Here are a few ideas you can think of to publicize this idea during the week:
1. Put up posters with some corporate jokes (take care they do not offend anyone or any position in particular).
2. Place some blank chart papers on the walls with markers hanging by the side, inviting people to make humorous comments and share their feedback on the idea itself, ideas for popularizing this concept, etc.
3. Distribute pocket cards with some inspiring quotes featuring the 'sign & greet' signature. The same can be done at the canteen or lunchrooms with tent cards on tables.
4. Organize competitions with a 'fun & greet' focus. For instance, 'greedy greeter' for the one who has greeted the most, hilarious greeter, humble greeter, gregarious greeter, etc.
I have just thrown out some ideas off the top of my head; probably at least one or two could be practicable in your context.
Regards,
BharathG