The answer to this is yes. You can always try to start the process and call an attorney later if you feel the need. This might not be advisable for a number of reasons, but if you need one, late is better than never. That said, there will be times where most attorneys will refuse to jump into the case. Taking on the responsibility of representing someone is something that we take seriously. Coming into a case midway means a lot of catchup for an attorney who has not been involved with the previous proceedings. They will need to know the history of the case by, at the very least, acquiring a copy of the court file and speaking with the opposing party's representation. Sometimes we will see the need to collect further evidence, which can take time—not atypically, months of time.
If you think you need an attorney, the very latest a prudent person should get counsel is at least a month before any planned trial. Depending on the circumstances, some attorneys may still refuse to get involved at this late date. We want to be prepared and maintain our reputation in doing so. If we feel we cannot meet those standards in a certain time frame, we may decline representation. We are also ethically obligated to provide diligent, competent services and shortening the timeline makes it that much harder to do so. On the other hand, if we think it is likely we can get a continuance of the trial date, we may jump into the case at a late date. Some courtrooms are more likely to do this than others. This has to do with the judge but also has to do with the docketing system. Some judges simply cannot afford to move a trial because their schedule or "docket" would be so backed up, cases would be delayed for an unduly long period of time.