Halliburton Co. sweetened its final offer only slightly from its first bid for rival Baker Hughes Inc., but the bump was magnified by a drop in Baker Hughes’s share price during merger talks this autumn between the world’s second- and third-largest oil-field service companies.

Halliburton on Monday disclosed details of their sometimes-tense courtship, which culminated in an agreement last month for Halliburton to acquire its smaller rival for cash and stock that valued Baker Hughes at around $35 billion.

The document reveals Halliburton’s initial offer of $19 and 1.05 of its own shares for each Baker Hughes share. That put a price tag of about $32.9 billion, or about $76 a share, on Baker Hughes, based on the closing price of Halliburton’s stock on Oct. 10, the last trading day before the Oct. 13 bid.

In the week that followed Halliburton’s initial overture, both companies reported third-quarter earnings. Halliburton beat expectations. Baker Hughes fell short. With Halliburton’s shares moving up, and Baker Hughes’s down, Halliburton’s offer effectively increased.

Halliburton emphasized the shift in negotiations. “Despite Baker Hughes’ earnings that surprised investors, we reaffirmed our proposal, and the premium has now increased to 45%,” Halliburton Chief Executive David Lesar wrote to his counterpart, Martin Craighead, on Nov. 3.

Meanwhile, Halliburton went ahead with plans to nominate a slate of directors to steer its rival into a merger. The day after Halliburton made its bid, for example, the company bought 100 Baker Hughes shares so that it could nominate directors for Baker Hughes’ April shareholder meeting, the filing said.

Halliburton and Baker Hughes compete for oil-field services business around the world, and the two companies dominate the market for some of their offerings. Baker Hughes’s board flagged antitrust risk as an issue from the outset.

Baker Hughes’s top legal officer proposed a $5 billion breakup fee if the deal was blocked by regulators and $3 billion if Halliburton’s shareholders voted down the deal. Baker Hughes also wanted Halliburton to commit to a “hell or high water” covenant—an agreement that Halliburton take any action necessary to get the deal past antitrust authorities.

Halliburton countered with a $1.5 billion breakup fee and a commitment to sell businesses that generated sales of several billion dollars if regulators required that.

The companies eventually met in the middle. Halliburton agreed to pay Baker Hughes $3.5 billion if regulators quashed the deal, and each agreed to smaller breakup fees if their shareholders vetoed the combination.

But they still hadn’t agreed on price. By Friday, Nov. 14, Halliburton delivered to Baker a slate of board candidates. Baker Hughes fired back with an after-hours news release chastising Halliburton’s tactics.

Halliburton persisted, with Mr. Lesar inviting Mr. Craighead to meet.

The next morning the Baker Hughes board met, and Mr. Craighead then reached out to Mr. Lesar. The CEOs met, with one lieutenant each at their sides. Mr. Lesar presented the final offer: $19 in cash and 1.12 of its own shares per Baker share, a roughly $2 billion bump that gave Baker Hughes shareholders a bigger stake in the combined company. The offer premium expanded from around 34% initially to more than 50% based on share prices a few days before, when The Wall Street Journal broke news of the talks.

It was enough for Baker Hughes, and the two sides spent the rest of the weekend working out the details.

Write to Ryan Dezember at ryan.dezember@wsj.com and Alison Sider at alison.sider@wsj.com