Here is what "Korean grammar for International learners" says (page 357):
The verb 싶다 can follow a wide variety of endings, including -나, -(으)ㄹ까, -(으)ㄹ성, -구나, -지, etc. In each case it expresses the speaker's guess or supposition, or else the speaker's judgement about some fact.
음식이 맛이 있나 싶어 먹어 보았습니다 = I ate the food thinking it might taste good.
오늘은 친구한테서 전화가 오지 않을까 싶다 = I have a feeling my friend might call today.
그 사람이 화를 낼 성싶어서 그 말을 하지 않았어요 = I thought he might get angry so I didn't say anything
그런 예쁜 여자도 있구나 싶었다 = I thought to myself 'Hey, there are girls as pretty as that, too?' (In other words, 'I had no idea there were girls as pretty as that!')
아마 그 친구는 학교에 갔지 싶어요 = She must certainly have gone to school (I reckon).